The Database of Integer Sequences, Part 5 

     Part of the On-Line Encyclopedia of Integer Sequences


This is a section of the main database for the On-Line Encyclopedia of Integer Sequences.

For more information see the following pages:
( www.research.att.com/~njas/sequences/ then )

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(start)



%I A096419
%S A096419 1,0,0,1,0,0,2,1,0,2,1,0,4,3,0,5,4,0,8,8,0,10,11,0,15,19,1,20,27,1,28,
%T A096419 43,3,36,61,6,50,92,11,64,129,18,86,189,33,110,262,51,145,374,84,184,
%U A096419 514,129,238,718,201,300,977,300,384,1344,454,482,1812,661,609,2459,972
%N A096419 Number of cyclically symmetric plane partitions (Macdonald's plane partition conjecture).
%C A096419 Equals A048141 (C3v symmetry) + 2* A048142 (only C3 symmetry).
%D A096419 Andrews, G. E. "Plane Partitions (III): The Weak Macdonald Conjecture." Invent. Math. 53, 193-225, 1979.
%D A096419 Mills, W. H.; Robbins, D. P.; and Rumsey, H. Jr., Proof of the Macdonald Conjecture. Invent. Math. 66, 73-87, 1982.
%H A096419 Wouter Meeussen, <a href="http://www.research.att.com/~njas/sequences/b096419.txt">Table of n, a(n) for n=1..151</a>
%H A096419 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MacdonaldsPlanePartitionConjecture.html">Macdonald's Plane Partition Conjecture</a>
%F A096419 See Mma code for a formula.
%t A096419 mcdon=Rest@CoefficientList[Series[Product[(1-q^(3i-1))/(1-q^(3i-2)) Product[(1-q^(3(m+i+j-1)))/(1-q^(3(2i+j-1))), {j, i, m}], {i, 1, m}]/.m->50, {q, 0, 97}], q]
%Y A096419 Cf. A047993, A048141, A048142.
%Y A096419 Adjacent sequences: A096416 A096417 A096418 this_sequence A096420 A096421 A096422
%Y A096419 Sequence in context: A025841 A138468 A029296 this_sequence A130182 A024361 A135486
%K A096419 nonn
%O A096419 1,7
%A A096419 Wouter Meeussen (wouter.meeussen(AT)pandora.be), Aug 08 2004

%I A130182
%S A130182 1,2,1,0,2,1,0,12,4,1,0,144,28,20,1,0,2880,216,508,50,1,0,86400,2592,17400,2548,
%T A130182 98,1,0,3628800,449280,788688,153760,8568,168,1,0,203212800,42405120,46032768,
%U A130182 11269008,811648,23016,264,1,0,14631321600,4187635200,3372731136
%V A130182 1,-2,1,0,-2,1,0,-12,4,1,0,-144,28,20,1,0,-2880,216,508,50,1,0,-86400,-2592,17400,2548,
%W A130182 98,1,0,-3628800,-449280,788688,153760,8568,168,1,0,-203212800,-42405120,46032768,
%X A130182 11269008,811648,23016,264,1,0,-14631321600,-4187635200,3372731136
%N A130182 Coefficients of the v=1 member of a family of certain orthogonal polynomials.
%C A130182 For v>=1 the orthogonal polynomials pt(n,v,x) have v integer zeros k*(k+1), k=1..v, for every n>=v and some other n-v zeros. The integer zeros are from 2*A000217.
%C A130182 The v-family pt(n,v,x) consists of characteristic polynomials of the tridiagonal M x M matrix Vt=Vt(M,v) with entries Vt_{m,n} given by 2*m*(v+1-m) if n=m, m=1,...,M; -m*(v+1-m) if n=m-1, m=2,...,M; -m*(v+1-m) if n=m+1, m=1..M-1 and 0 else. pt(n,v,x):=det(x*I_n-Vt(n,v) with the n dimensional unit matrix I_n.
%C A130182 pt(n,v=1,x) has, for every n>=1, among its n zeros one for x=2. pt(1,1,x) has therefore only the integer zeros 2. det(Vt(1,1))=2.
%C A130182 The column sequences give [1,-2,0,0,0,...], A010790(n-1)*(-1)^(n-1), A130185, A130186 for m=0,1,2,3.
%C A130182 Coefficients of pt(n,v=1,x) (in the quoted Bruschi et al. paper {\tilde p}^{(\nu)}_n(x) of eqs. (20) and (24a),(24b)) in increasing powers of x.
%D A130182 M. Bruschi, F. Calogero and R. Droghei, Proof of certain Diophantine conjectures and identification of remarkable classes of orthogonal polynomials, J. Physics A, 40(2007)3815-3829.
%H A130182 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A130182.text">First ten rows and more. </a>
%F A130182 a(n,m)=[x^m]pt(n,1,x), n>=0, with the three term recurrence for orthogonal polynomial systems of the form pt(n,v,x) = (x + 2*n*(n-1-v)*pt(n-1,v,x) -(n-1)*n*(n-1-v)*(n-2-v)*pt(n-2,v,x), n>=1; pt(-1,v,x)=0 and pt(0,v,x)=1. Put v=1 here.
%F A130182 Recurrence: a(n,m) = a(n-1,m-1)+2*n*(n-2)*a(n-1,m) - (n-1)*n*(n-2)*(n-3)*a(n-2,m); a(n,m)=0 if n<m, a(-1,m):=0, a(n,-1)=1. From the pt(n,1,x) recurrence.
%e A130182 [1]; [ -2,1]; [0,-2,1]; [0,-12,4,1]; [0,-144,28,20,1]; [0,-2880,216,508,50,1]; ...
%e A130182 Row n=5:[0,-2880,216,508,50,1]; pt(5,2,x)= x*(-2880+216*x+508*x^2+50*x^3+1*x^4)= x*(x-2)*(1440+612*x+52*x^2+x^3). pt(5,1,x) has the guaranteed integer zero x=2 (and also x=0 and some other three zeros).
%e A130182 Row n=1:[ -2,1]. pt(1,1,x)=-2+x with integer zero x=2.
%Y A130182 Cf. A129065 (a v=1 member of a similar family).
%Y A130182 Row sums A130183, unsigned row sums A130184.
%Y A130182 Adjacent sequences: A130179 A130180 A130181 this_sequence A130183 A130184 A130185
%Y A130182 Sequence in context: A138468 A029296 A096419 this_sequence A024361 A135486 A030187
%K A130182 sign,tabl,easy
%O A130182 0,2
%A A130182 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jun 01 2007

%I A024361
%S A024361 0,0,1,1,1,0,1,1,1,0,1,2,1,0,2,1,1,0,1,2,2,0,1,2,1,0,1,2,1,0,1,1,2,
%T A024361 0,2,2,1,0,2,2,1,0,1,2,2,0,1,2,1,0,2,2,1,0,2,2,2,0,1,4,1,0,2,1,2,0,
%U A024361 1,2,2,0,1,2,1,0,2,2,2,0,1,2,1,0,1,4,2,0,2,2,1,0,2,2,2,0,2,2,1,0,2
%N A024361 Number of primitive Pythagorean triangles with leg n.
%C A024361 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives number of times AUB takes value n.
%C A024361 For n>1, a(n)=0 for n=A016825=2(mod 4). Also, number of ways of expressing n as a difference of two coprime squares. - Lekraj Beedassy (blekraj(AT)yahoo.com), Sep 28 2004
%H A024361 Ron Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>
%H A024361 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PythagoreanTriple.html">Link to a section of The World of Mathematics.</a>
%Y A024361 Cf. A024362, A046079.
%Y A024361 Cf. A020883; A020884.
%Y A024361 Adjacent sequences: A024358 A024359 A024360 this_sequence A024362 A024363 A024364
%Y A024361 Sequence in context: A029296 A096419 A130182 this_sequence A135486 A030187 A117278
%K A024361 nonn
%O A024361 1,12
%A A024361 David W. Wilson (davidwwilson(AT)comcast.net)

%I A135486
%S A135486 1,0,2,1,0,2,1,1,0,3,5,0,0,0,2,2,4,1,0,0,4,13,0,0,0,0,0,2,8,7,0,3,0,0,0,
%T A135486 4,21,0,6,0,0,0,0,0,3,17,20,0,0,1,0,0,0,0,4,54,0,0,0,0,0,0,0,0,0,2,31,
%U A135486 18,6,10,0,6,0,0,0,0,0,6,99,0,0,0,0,0,0,0,0,0,0,0,2,70,60,0,0,0,0,1,0,0
%N A135486 Triangle read by rows: T(n,k) = number of partitions of n having k-fold symmetry, cf. A085436.
%e A135486 1; 0,2; 1,0,2; 1,1,0,3; 5,0,0,0,2; 2,4,1,0,0,4; 13,0,0,0,0,0,2; 8,7,0,3,0,0,0,4; ...
%Y A135486 Adjacent sequences: A135483 A135484 A135485 this_sequence A135487 A135488 A135489
%Y A135486 Sequence in context: A096419 A130182 A024361 this_sequence A030187 A117278 A140082
%K A135486 nonn,tabl
%O A135486 1,3
%A A135486 Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 07 2008

%I A030187
%S A030187 1,1,2,1,0,2,1,1,1,0,0,2,4,1,0,1,6,1,2,0,2,0,0,2,5,4,4,1,6,0,4,1,0,6,0,
%T A030187 1,2,2,8,0,6,2,8,0,0,0,12,2,1,5,12,4,6,4,0,1,4,6,6,0,8,4,1,1,0,0,4,6,0,
%U A030187 0,0,1,2,2,10,2,0,8,8,0,11,6,6,2,0,8,12,0,6,0,4,0,8,12,0,2,10,1,0,5,0
%V A030187 1,-1,-2,1,0,2,1,-1,1,0,0,-2,-4,-1,0,1,6,-1,2,0,-2,0,0,2,-5,4,4,1,-6,0,-4,-1,0,-6,0,1,
%W A030187 2,-2,8,0,6,2,8,0,0,0,-12,-2,1,5,-12,-4,6,-4,0,-1,-4,6,-6,0,8,4,1,1,0,0,-4,6,0,0,0,-1,
%X A030187 2,-2,10,2,0,-8,8,0,-11,-6,-6,-2,0,-8,12,0,-6,0,-4,0,8,12,0,2,-10,-1,0,-5,0
%N A030187 Expansion of eta(q)*eta(q^2)*eta(q^7)*eta(q^14) in powers of q.
%D A030187 M. Koike, Matheiu group M24 and modular forms, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)
%F A030187 Euler transform of period 14 sequence [ -1, -2, -1, -2, -1, -2, -2, -2, -1, -2, -1, -2, -1, -4, ...]. - Michael Somos Aug 13 2006
%F A030187 a(n) is multiplicative with a(2^e) = (-1)^e, a(7^e) = 1, otherwise a(p^e) = a(p)a(p^(e-1))-p*a(p^(e-2)) where a(p) = p minus number of points of elliptic curve modulo p . - Michael Somos Aug 13 2006
%F A030187 G.f.: x Product_{k>0} (1-x^k)(1-x^(2k))(1-x^(7k))(1-x^(14k)).
%F A030187 Coefficients of L-series for elliptic curve "14a4": y^2 +x*y +y= x^3 -x or y^2 +x*y -y= x^3 . - Michael Somos Feb 19 2007
%F A030187 G.f. A(x) satisfies 0= f(A(x), A(x^2), A(x^4)) where f(u, v, w)= v^4 -u*w* (u+2*v)* (v+2*w) . - Michael Somos Feb 19 2007
%F A030187 Associated with permutations in Mathieu group M24 of shape (14)(7)(2)(1).
%F A030187 G.f. is Fourier series of a weight 2 level 14 modular form. f(-1/ (14 t)) = 14 (t/i)^2 f(t) where q = exp(2 pi i t).
%o A030187 (PARI) {a(n)=if(n<1, 0, ellak(ellinit([ -1, 0, -1, -1, 0]), n))} /* Michael Somos Aug 13 2006 */
%o A030187 (PARI) {a(n)= local(A, p, e, x, y, a0, a1); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, (-1)^e, if(p==7, 1, a0=1; a1=y=-sum(x=0, p-1, kronecker(4*x^3+x^2-2*x+1, p)); for(i=2, e, x=y*a1-p*a0; a0=a1; a1=x); a1)))))} /* Michael Somos Aug 13 2006 */
%o A030187 (PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff(eta(x+A)*eta(x^2+A)*eta(x^7+A)*eta(x^14+A), n))}
%Y A030187 Adjacent sequences: A030184 A030185 A030186 this_sequence A030188 A030189 A030190
%Y A030187 Sequence in context: A130182 A024361 A135486 this_sequence A117278 A140082 A025852
%K A030187 sign,mult
%O A030187 1,3
%A A030187 njas

%I A117278
%S A117278 1,1,0,1,1,1,0,1,1,1,1,1,0,1,1,1,0,1,2,1,0,2,1,1,1,1,0,2,2,1,0,1,2,2,1,
%T A117278 1,1,1,2,2,2,1,0,2,1,3,2,1,1,0,1,3,2,3,2,1,0,2,2,3,3,2,1,1,1,0,4,3,3,3,
%U A117278 2,1,0,2,2,4,3,4,2,1,1,1,1,3,4,5,3,3,2,1,0,2,2,6,4,4,4,2,1,1,0,1,5,3,6
%N A117278 Triangle read by rows: T(n,k) is the number of partitions of n into k prime parts (n>=2, 1<=k<=floor(n/2)).
%C A117278 Row n has floor(n/2) terms. Row sums yield A000607. T(n,1)=A010051(n) (the characteristic function of the primes). T(n,2)=A061358(n). Sum(k*T(n,k),k>=1) = A084993(n).
%F A117278 G.f.=G(t,x)=-1+1/product(1-tx^(p(j)), j=1..infinity), where p(j) is the j-th prime.
%e A117278 T(12,3)=2 because we have [7,3,2] and [5,5,2].
%e A117278 Triangle starts:
%e A117278 1;
%e A117278 1;
%e A117278 0,1;
%e A117278 1,1;
%e A117278 0,1,1;
%e A117278 1,1,1;
%e A117278 0,1,1,1;
%e A117278 0,1,2,1;
%p A117278 g:=1/product(1-t*x^(ithprime(j)),j=1..30): gser:=simplify(series(g,x=0,30)): for n from 2 to 22 do P[n]:=sort(coeff(gser,x^n)) od: for n from 2 to 22 do seq(coeff(P[n],t^j),j=1..floor(n/2)) od; # yields sequence in triangular form
%Y A117278 Cf. A000607, A010051, A061358, A084993.
%Y A117278 Adjacent sequences: A117275 A117276 A117277 this_sequence A117279 A117280 A117281
%Y A117278 Sequence in context: A024361 A135486 A030187 this_sequence A140082 A025852 A025846
%K A117278 nonn,tabf
%O A117278 2,19
%A A117278 Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 07 2006

%I A140082
%S A140082 0,1,1,2,1,0,2,1,1,2,0,1,2,1,1,0,1,2,2,3,0,1,1,2,2,0,1,1,1,1,0,1,1,
%T A140082 2,2,0,2,1,3,1,0,1,1,1,1,0,2,1,2,3,0,1,1,2,1,0,1,1,1,2,0,1,1,1,1,0,
%U A140082 2,1,2,1,0,1,2,1,1,0,3,2,1,1,0,1,1,2,1,0,1,1,1,1,0,1,2,1,1,0,2,1,3
%N A140082 Same as A140080, except now e=5.
%H A140082 Nadia Heninger and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b140082.txt">Table of n, a(n) for n = 0..5000</a>
%o A140082 See link in A140080 for Fortran program.
%Y A140082 Adjacent sequences: A140079 A140080 A140081 this_sequence A140083 A140084 A140085
%Y A140082 Sequence in context: A135486 A030187 A117278 this_sequence A025852 A025846 A033780
%K A140082 nonn
%O A140082 0,4
%A A140082 Nadia Heninger (nadiah(AT)cs.princeton.edu) and njas, Jun 03 2008

%I A025852
%S A025852 1,0,0,1,0,0,1,0,1,1,0,2,1,0,2,1,1,2,1,2,2,1,3,2,2,3,2,
%T A025852 3,3,2,4,3,3,5,3,4,5,3,5,5,4,6,5,5,7,5,6,7,6,7,7,7,8,7,
%U A025852 8,9,8,9,9,9,10,9,10,11,10,11,12,11,12,12,12,13,13,13,14
%N A025852 Expansion of 1/((1-x^3)(1-x^8)(1-x^11)).
%Y A025852 Adjacent sequences: A025849 A025850 A025851 this_sequence A025853 A025854 A025855
%Y A025852 Sequence in context: A030187 A117278 A140082 this_sequence A025846 A033780 A035210
%K A025852 nonn
%O A025852 0,12
%A A025852 njas

%I A025846
%S A025846 1,0,0,1,0,0,1,1,0,2,1,0,2,1,1,2,2,1,3,2,1,4,2,2,4,3,2,
%T A025846 5,4,2,6,4,3,6,5,4,7,6,4,8,6,5,9,7,6,10,8,6,11,9,7,12,10,
%U A025846 8,13,11,9,14,12,10,15,13,11,17,14,12,18,15,13,19,17,14
%N A025846 Expansion of 1/((1-x^3)(1-x^7)(1-x^9)).
%Y A025846 Adjacent sequences: A025843 A025844 A025845 this_sequence A025847 A025848 A025849
%Y A025846 Sequence in context: A117278 A140082 A025852 this_sequence A033780 A035210 A029295
%K A025846 nonn
%O A025846 0,10
%A A025846 njas

%I A033780
%S A033780 1,1,0,2,1,0,2,1,1,2,4,1,1,3,0,2,2,2,2,3,1,4,4,0,7,4,2,
%T A033780 4,5,1,6,11,1,4,4,2,6,5,4,6,8,1,6,6,1,14,8,3,6,6,0
%N A033780 Product t2(q^d); d | 21, where t2 = theta2(q)/(2*q^(1/4)).
%Y A033780 Adjacent sequences: A033777 A033778 A033779 this_sequence A033781 A033782 A033783
%Y A033780 Sequence in context: A140082 A025852 A025846 this_sequence A035210 A029295 A130094
%K A033780 nonn
%O A033780 0,4
%A A033780 njas

%I A035210
%S A035210 1,1,2,1,0,2,1,1,3,0,0,2,0,1,0,1,0,3,2,0,2,0,0,2,1,0,4,
%T A035210 1,2,0,2,1,0,0,0,3,2,2,0,0,0,2,0,0,0,0,2,2,1,1,0,0,2,4,
%U A035210 0,1,4,2,2,0,0,2,3,1,0,0,0,0,0,0,0,3,0,2,2,2,0,0,0,0,5
%N A035210 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 28.
%o A035210 (PARI) direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
%Y A035210 Adjacent sequences: A035207 A035208 A035209 this_sequence A035211 A035212 A035213
%Y A035210 Sequence in context: A025852 A025846 A033780 this_sequence A029295 A130094 A130210
%K A035210 nonn
%O A035210 1,3
%A A035210 njas

%I A029295
%S A029295 1,0,0,1,0,0,2,1,0,2,1,1,3,2,2,3,2,3,5,3,4,6,4,5,8,6,6,
%T A029295 9,8,8,11,10,10,13,12,13,16,14,15,19,17,18,23,20,21,26,
%U A029295 24,25,30,28,29,34,32,34,39,37,39,44,42,44,50,48,50,56
%N A029295 Expansion of 1/((1-x^3)(1-x^6)(1-x^7)(1-x^11)).
%Y A029295 Adjacent sequences: A029292 A029293 A029294 this_sequence A029296 A029297 A029298
%Y A029295 Sequence in context: A025846 A033780 A035210 this_sequence A130094 A130210 A035443
%K A029295 nonn
%O A029295 0,7
%A A029295 njas

%I A130094
%S A130094 1,1,2,1,0,2,1,2,0,1,1,0,0,0,2,1,2,2,0,0,4,1,0,0,0,0,0,2,1,2,0,1,0,0,0,
%T A130094 0,1,0,2,0,0,0,0,0,1,1,2,0,0,2,0,0,0,0,4
%V A130094 1,1,-2,1,0,-2,1,-2,0,1,1,0,0,0,-2,1,-2,-2,0,0,4,1,0,0,0,0,0,-2,1,-2,0,1,0,0,0,0,1,0,
%W A130094 -2,0,0,0,0,0,1,1,-2,0,0,-2,0,0,0,0,4
%N A130094 A051731 * an infinite lower triangular matrix with A007427 in the mail diagonal.
%C A130094 Right border = A007427. Row sums = mu(n), A008683. Nonzero terms by rows are substituted for the factors of n such that row sums = mu(n). Example: for row 6 we map (1, -2, -2, 0, 4), sum = 1; since the factors of 6 are 1, 2, 3, and 6.
%F A130094 Inverse Moebius transform of a triangular matrix with A007427 in the main diagonal and the rest zeros.
%e A130094 First few rows of the triangle are:
%e A130094 1;
%e A130094 1, -2;
%e A130094 1, 0, -2;
%e A130094 1, -2, 0, 1;
%e A130094 1, 0, 0, 0, -2;
%e A130094 1, -2, -2, 0, 0, 4;
%e A130094 1, 0, 0, 0, 0, 0, -2;
%e A130094 1, -2, 0, 1, 0, 0, 0, 0;
%e A130094 1, 0, -2, 0, 0, 0, 0, 0, 1;
%e A130094 ...
%Y A130094 Cf. A007427, A051731, A008683.
%Y A130094 Adjacent sequences: A130091 A130092 A130093 this_sequence A130095 A130096 A130097
%Y A130094 Sequence in context: A033780 A035210 A029295 this_sequence A130210 A035443 A036261
%K A130094 tabl,sign
%O A130094 1,3
%A A130094 Gary W. Adamson (qntmpkt(AT)yahoo.com), May 06 2007

%I A130210
%S A130210 1,1,2,1,0,2,1,2,0,3,1,0,0,0,2,1,2,2,0,0,4,1,0,0,0,0,0,2,1,2,0,3,0,0,0,
%T A130210 4,1,0,2,0,0,0,0,0,3,1,2,0,0,2,0,0,0,0,4
%N A130210 A051731 * A130209.
%C A130210 Right border = A000005, d(n). Row sums = A007425: (1, 3, 3, 6, 3, 9, 3,...).
%F A130210 Inverse Moebius transform of A130209
%e A130210 First few rows of the triangle are:
%e A130210 1;
%e A130210 1, 2;
%e A130210 1, 0, 2;
%e A130210 1, 2, 0, 3;
%e A130210 1, 0, 0, 0, 2;
%e A130210 1, 2, 2, 0, 0, 4;
%e A130210 1, 0, 0, 0, 0, 0, 2;
%e A130210 ...
%Y A130210 Cf. A000005, A007425.
%Y A130210 Adjacent sequences: A130207 A130208 A130209 this_sequence A130211 A130212 A130213
%Y A130210 Sequence in context: A035210 A029295 A130094 this_sequence A035443 A036261 A091917
%K A130210 nonn,tabl
%O A130210 0,3
%A A130210 Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2007

%I A035443
%S A035443 0,0,1,0,0,1,0,1,1,0,2,1,0,2,1,2,2,1,4,2,1,5,2,4,5,2,8,5,2,10,5,7,11,5,
%T A035443 14,11,5,19,11,12,21,11,24,22,11,33,22,22,38,22,41,40,22,58,41,37,68,
%U A035443 41,67,73,41,95,75,63,114,76,108,124,76,155,129,106,188,131,173
%N A035443 Number of partitions of n into parts 8k or 8k+3.
%Y A035443 Adjacent sequences: A035440 A035441 A035442 this_sequence A035444 A035445 A035446
%Y A035443 Sequence in context: A029295 A130094 A130210 this_sequence A036261 A091917 A025657
%K A035443 nonn
%O A035443 1,11
%A A035443 Olivier Gerard (olivier.gerard(AT)gmail.com)

%I A036261
%S A036261 1,1,2,1,0,2,1,2,2,4,1,2,0,2,2,1,2,0,0,2,4,1,2,0,0,0,2,2,1,2,0,0,0,0,2,
%T A036261 4,1,2,0,0,0,0,0,2,6,1,0,2,2,2,2,2,2,4,2,1,0,0,2,0,2,0,2,0,4,6,1,0,0,0,
%U A036261 2,2,0,0,2,2,2,4,1,0,0,0,0,2,0,0,0,2,0,2,2,1,0,0,0,0,0,2,2,2,2,0,0,2,4
%N A036261 Triangle of numbers arising from Gilbreath's conjecture: successive absolute differences of primes.
%D A036261 R. K. Guy, Unsolved Problems Number Theory, A10.
%H A036261 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b036261.txt">Rows n=1..100 of triangle, flattened</a>
%H A036261 A. M. Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/doc/cnt.html">Iterated absolute values of differences of consecutive primes</a>, Math. Comp. 61 (1993), 373-380.
%Y A036261 See A036262, which is the main entry for this array.
%Y A036261 Adjacent sequences: A036258 A036259 A036260 this_sequence A036262 A036263 A036264
%Y A036261 Sequence in context: A130094 A130210 A035443 this_sequence A091917 A025657 A025686
%K A036261 tabl,easy,nice,nonn
%O A036261 1,3
%A A036261 njas
%E A036261 More terms from Naohiro Nomoto (6284968128(AT)geocities.co.jp), May 22 2001

%I A091917
%S A091917 1,2,1,0,2,1,2,3,3,1,0,4,6,4,1,2,5,10,10,5,1,0,6,15,20,15,6,1,2,7,21,35,
%T A091917 35,21,7,1,0,8,28,56,70,56,28,8,1,2,9,36,84,126,126,84,36,9,1,0,10,45,
%U A091917 120,210,252,210,120,45,10,1,2,11,55,165,330,462,462,330,165
%V A091917 1,-2,1,0,-2,1,-2,3,-3,1,0,-4,6,-4,1,-2,5,-10,10,-5,1,0,-6,15,-20,15,-6,1,-2,7,-21,35,
%W A091917 -35,21,-7,1,0,-8,28,-56,70,-56,28,-8,1,-2,9,-36,84,-126,126,-84,36,-9,1,0,-10,45,-120,
%X A091917 210,-252,210,-120,45,-10,1,-2,11,-55,165,-330,462,-462,330,-165
%N A091917 Coefficient array of polynomials (z-1)^n-1.
%C A091917 The first element has been changed to 1 to produce an invertible matrix. Alternatively, this is the coefficient array for the polynomials P(z,n)= product{j=0..n-1, z-(1+w(n)^j)} where w(n)=e^(2pi*i/n), i=sqrt(-1).
%C A091917 The row entries determine interesting recurrences. For instance, a(n)=4a(n-1)+6a(n-2)+4a(n-3), a(0)=a(1)=a(2)=1, gives A038503. Sequences of the form a(n)=sum{k=0..n, if (mod(k,m)=r, binomial(n,k), 0)}, for r=0..m-1, result. Equivalently, a(n)=sum{j=0..n-1, 2^n(cos(pi*j/m))^n*cos((n-2r)pi*j/m)}/m, r=0..m-1. These include A024493, A024494, A024495, A038503, A038504, A038505. The inverse matrix is A091918.
%C A091917 Triangle T(n,k), 0<=k<=n, read by rows given by [ -2, 2, 1/2, -1/2, 0, 0, 0, 0, 0, ...] DELTA [1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 11 2007
%e A091917 Rows begin {1}, {-2,1}, {0,-2,1}, {-2, 3, -3, 1}, {0,-4, 6, -4, 1},...
%Y A091917 Adjacent sequences: A091914 A091915 A091916 this_sequence A091918 A091919 A091920
%Y A091917 Sequence in context: A130210 A035443 A036261 this_sequence A025657 A025686 A022329
%K A091917 sign,tabl
%O A091917 0,2
%A A091917 Paul Barry (pbarry(AT)wit.ie), Feb 13 2004

%I A025657
%S A025657 0,0,1,0,1,0,2,1,0,2,1,3,0,2,1,3,0,2,4,1,3,0,2,4,1,3,0,5,2,4,1,3,0,5,2,4,
%T A025657 1,6,3,0,5,2,4,1,6,3,0,5,2,7,4,1,6,3,0,5,2,7,4,1,6,3,0,8,5,2,7,4,1,6,3,0,
%U A025657 8,5,2,7,4,1,9,6,3,0,8,5,2,7,4,1,9,6,3,0,8,5,2,10,7,4,1,9,6,3,0,8,5,2,10
%N A025657 Exponent of 6 (value of j) in n-th number of form 3^i*6^j.
%Y A025657 Differs from A025686 at a(114).
%Y A025657 Adjacent sequences: A025654 A025655 A025656 this_sequence A025658 A025659 A025660
%Y A025657 Sequence in context: A035443 A036261 A091917 this_sequence A025686 A022329 A087466
%K A025657 nonn
%O A025657 1,7
%A A025657 David W. Wilson (davidwwilson(AT)comcast.net)

%I A025686
%S A025686 0,0,1,0,1,0,2,1,0,2,1,3,0,2,1,3,0,2,4,1,3,0,2,4,1,3,0,5,2,4,1,3,0,5,2,4,
%T A025686 1,6,3,0,5,2,4,1,6,3,0,5,2,7,4,1,6,3,0,5,2,7,4,1,6,3,0,8,5,2,7,4,1,6,3,0,
%U A025686 8,5,2,7,4,1,9,6,3,0,8,5,2,7,4,1,9,6,3,0,8,5,2,10,7,4,1,9,6,3,0,8,5,2,10
%N A025686 Exponent of 10 (value of j) in n-th number of form 4^i*10^j.
%Y A025686 Differs from A025657 at a(114).
%Y A025686 Adjacent sequences: A025683 A025684 A025685 this_sequence A025687 A025688 A025689
%Y A025686 Sequence in context: A036261 A091917 A025657 this_sequence A022329 A087466 A071432
%K A025686 nonn
%O A025686 1,7
%A A025686 David W. Wilson (davidwwilson(AT)comcast.net)

%I A022329
%S A022329 0,0,1,0,1,0,2,1,0,2,1,3,0,2,1,3,0,2,4,1,3,0,2,4,1,3,5,0,2,4,1,3,5,0,2,4,
%T A022329 6,1,3,5,0,2,4,6,1,3,5,0,7,2,4,6,1,3,5,0,7,2,4,6,1,8,3,5,0,7,2,4,6,1,8,3,
%U A022329 5,0,7,2,9,4,6,1,8,3,5,0,7,2,9,4,6,1,8,3,10,5,0,7,2,9,4,6,1,8,3,10,5,0,7
%N A022329 Exponent of 3 (value of j) in n-th number of form 2^i*3^j.
%Y A022329 Adjacent sequences: A022326 A022327 A022328 this_sequence A022330 A022331 A022332
%Y A022329 Sequence in context: A091917 A025657 A025686 this_sequence A087466 A071432 A025253
%K A022329 nonn
%O A022329 1,7
%A A022329 Clark Kimberling (ck6(AT)evansville.edu)

%I A087466
%S A087466 0,0,1,0,1,0,2,1,0,2,1,3,0,2,1,3,0,2,4,1,3,0,2,4,1,3,5,0,2,4,1,3,5,0,2,
%T A087466 4,6,1,3,5,0,2,4,6,1,3,5,7,0,2,4,6,1,3,5,7,0,2,4,6,8,1,3,5,7,0,2,4,6,8,
%U A087466 1,3,5,7,9,0,2,4,6,8,1,3,5,7,9,0,2,4,6,8,10,1,3,5,7,9,0,2,4,6,8,10,1,3
%N A087466 a(n) = number of the row (counting from initial row 0) of the array R in A087465 that contains n.
%C A087466 A sequence that contains itself as a proper subsequence (infinitely many times); that is, a fractal sequence.
%H A087466 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%e A087466 Northwest corner of R:
%e A087466 1 2 4 6 9
%e A087466 3 5 8 11 15
%e A087466 7 10 14 18 23
%e A087466 12 16 21 26 32
%e A087466 19 24 30 36 43
%e A087466 a(10)=2 because 10 is in row 2.
%Y A087466 Cf. A087465, A087467.
%Y A087466 Adjacent sequences: A087463 A087464 A087465 this_sequence A087467 A087468 A087469
%Y A087466 Sequence in context: A025657 A025686 A022329 this_sequence A071432 A025253 A112178
%K A087466 nonn
%O A087466 1,7
%A A087466 Clark Kimberling (ck6(AT)evansville.edu), Sep 09 2003

%I A071432
%S A071432 1,0,2,1,0,2,1,3,2,1,3,2,4,3,0,4,3,2,4,1,2,3,1,2,0,1,2,4,1,5,4,1,5,4,1,
%T A071432 5,2,1,0,2,1,3,2,1,3,2,4,3,2,4,3,2,4,3,2,4,3,2,0,3,2,4,8,5,4,3,5,4,6,5,
%U A071432 2,6,5,2,1,3,2,1,3,2,4,3,2,4,3,2,4,3,2,4,3,2,0,3,2,4,3,5,4,3,5,4
%N A071432 Sprague-Grundy values for octal games .36 and .361.
%D A071432 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982; see Chapter 4, p. 106.
%H A071432 Achim Flammenkamp, <a href="http://www.uni-bielefeld.de/~achim/octal_sparse.html">Octal games</a>
%Y A071432 Adjacent sequences: A071429 A071430 A071431 this_sequence A071433 A071434 A071435
%Y A071432 Sequence in context: A025686 A022329 A087466 this_sequence A025253 A112178 A134663
%K A071432 nonn
%O A071432 1,3
%A A071432 njas and Sue Pope (pope(AT)research.att.com), May 29 2002
%E A071432 Edited and extended by Christian G. Bower (bowerc(AT)usa.net), Oct 22 2002

%I A025253
%S A025253 0,2,1,0,2,1,4,6,10,24,36,85,152,310,638,1232,2620,5194,10840,22332,46004,
%T A025253 96528,199816,420365,880112,1851366,3908854,8245536,17482220,37049662,
%U A025253 78746776,167602164,357127116,762506640,1629288824,3487128706,7470956240
%N A025253 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-3)*a(3) for n >= 4.
%Y A025253 Adjacent sequences: A025250 A025251 A025252 this_sequence A025254 A025255 A025256
%Y A025253 Sequence in context: A022329 A087466 A071432 this_sequence A112178 A134663 A000925
%K A025253 nonn
%O A025253 1,2
%A A025253 Clark Kimberling (ck6(AT)evansville.edu)

%I A112178
%S A112178 1,1,0,1,1,0,0,1,0,1,0,0,1,0,0,1,1,0,2,1,0,2,2,0,0,1,0,2,1,0,2,1,0,1,2,
%T A112178 0,4,3,0,4,3,0,0,3,0,5,2,0,4,2,0,2,3,0,8,5,0,7,6,0,1,5,0,9,4,0,8,4,0,3,
%U A112178 6,0,14,9,0,13,10,0,2,9,0,16,8,0,14,8,0,5,11,0,24,14,0,21,16,0,3
%V A112178 1,-1,0,-1,-1,0,0,-1,0,1,0,0,-1,0,0,-1,-1,0,2,-1,0,-2,-2,0,0,-1,0,2,-1,0,-2,-1,0,-1,-2,
%W A112178 0,4,-3,0,-4,-3,0,0,-3,0,5,-2,0,-4,-2,0,-2,-3,0,8,-5,0,-7,-6,0,-1,-5,0,9,-4,0,-8,-4,0,
%X A112178 -3,-6,0,14,-9,0,-13,-10,0,-2,-9,0,16,-8,0,-14,-8,0,-5,-11,0,24,-14,0,-21,-16,0,-3
%N A112178 McKay-Thompson series of class 36i for the Monster group.
%D A112178 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%e A112178 T36i = 1/q -q -q^5 -q^7 -q^13 +q^17 -q^23 -q^29 -q^31 +2*q^35 +...
%Y A112178 Adjacent sequences: A112175 A112176 A112177 this_sequence A112179 A112180 A112181
%Y A112178 Sequence in context: A087466 A071432 A025253 this_sequence A134663 A000925 A003985
%K A112178 sign
%O A112178 0,19
%A A112178 Michael Somos, Aug 28 2005

%I A134663
%S A134663 1,1,1,0,0,1,1,2,1,0,2,2,0,0,1,1,3,3,1,3,6,3,0,3,3,0,0,1,1,4,6,4,5,12,
%T A134663 12,4,6,12,6,0,4,4,0,0,1,1,5,10,10,10,21,30,20,15,30,30,10,10,20,10,0,5,
%U A134663 5,0,0,1,1,6,15,20,21,36,61,60,45,66,90,60,35,60,60,20,15,30,15,0,6,6,0
%N A134663 Triangle read by rows in which n-th row gives the expansion coefficients of (1+x+x^4)^n.
%C A134663 The n-th row has 4n+1 entries.
%H A134663 S. Finch, P. Sebah and Z.-Q. Bai, <a href="http://arXiv.org/abs/0802.2654">Odd Entries in Pascal's Trinomial Triangle</a> (arXiv:0802.2654)
%t A134663 CoefficientList[(1+x+x^4)^n, x]
%Y A134663 Cf. A027907, A038717.
%Y A134663 Adjacent sequences: A134660 A134661 A134662 this_sequence A134664 A134665 A134666
%Y A134663 Sequence in context: A071432 A025253 A112178 this_sequence A000925 A003985 A065676
%K A134663 nonn,tabf
%O A134663 0,8
%A A134663 Steven Finch (Steven.Finch(AT)inria.fr), Jan 25 2008

%I A000925
%S A000925 1,2,1,0,2,2,0,0,1,2,2,0,0,2,0,0,2,2,1,0,2,0,0,0,0,4,2,0,0,2,0,0,1,0,2,0,2,2,0,
%T A000925 0,2,2,0,0,0,2,0,0,0,2,3,0,2,2,0,0,0,0,2,0,0,2,0,0,2,4,0,0,2,0,0,0,1,2,2,0,0,
%U A000925 0,0,0,2,2,2,0,0,4,0,0,0,2,2,0,0,0,0,0,0,2,1,0,4
%N A000925 Number of ordered ways of writing n as a sum of 2 squares of nonnegative integers.
%D A000925 A. Das and A. C. Melissinos, Quantum Mechanics: A Modern Introduction, Gordon and Breach, 1986, p. 47.
%D A000925 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985.
%H A000925 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000925.txt">Table of n, a(n) for n=0..10000</a>
%H A000925 <a href="http://www.research.att.com/~njas/sequences/Sindx_Su.html#ssq">Index entries for sequences related to sums of squares</a>
%F A000925 Coefficient of q^k in (1/4)*(1 + theta_3(0, q))^2.
%F A000925 a(A001481(n))>0; a(A022544(n))=0. - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 20 2003
%o A000925 (PARI) a(n)=sum(i=0,n,sum(j=0,n,if(i^2+j^2-n,0,1)))
%Y A000925 Adjacent sequences: A000922 A000923 A000924 this_sequence A000926 A000927 A000928
%Y A000925 Sequence in context: A025253 A112178 A134663 this_sequence A003985 A065676 A003263
%K A000925 nonn,nice
%O A000925 0,2
%A A000925 Jacques Haubrich (jhaubrich(AT)freeler.nl)

%I A003985
%S A003985 1,0,0,1,2,1,0,2,2,0,1,0,3,0,1,0,0,0,0,0,0,1,2,1,4,1,2,1,0,2,2,4,4,2,2,
%T A003985 0,1,0,3,4,5,4,3,0,1,0,0,0,4,4,4,4,0,0,0,1,2,1,0,5,6,5,0,1,2,1,0,2,2,0,
%U A003985 0,6,6,0,0,2,2,0,1,0,3,0,1,0,7,0,1,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A003985 Triangle with subscripts (1,1),(2,1),(1,2),(3,1),(2,2),(1,3), etc. in which entry (i,j) is i AND j.
%Y A003985 Antidiagonal sums are in A006581.
%Y A003985 Adjacent sequences: A003982 A003983 A003984 this_sequence A003986 A003987 A003988
%Y A003985 Sequence in context: A112178 A134663 A000925 this_sequence A065676 A003263 A135211
%K A003985 tabl,nonn,easy
%O A003985 1,5
%A A003985 Marc LeBrun (mlb(AT)well.com)
%E A003985 More terms from Larry Reeves (larryr(AT)acm.org), Sep 21 2000

%I A065676
%S A065676 0,1,1,0,1,1,0,2,1,0,2,2,0,1,2,0,1,3,0,2,3,0,2,2,0,3,2,0,3,1,0,1,1,0,1,2,
%T A065676 0,2,2,0,2,3,0,1,3,0,1,1,0,3,1,0,3,2,0,2,2,0,2,1,0,1,1,0,1,1,0,2,1,0,2,1,
%U A065676 0,3,1,0,3,4,0,1,4,0,1,1,0,2,1,0,2,1,0,1,1,0,1,1,0,1,1,0,1,2,0,1,2,0,1,1
%V A065676 0,-1,1,0,1,-1,0,-2,1,0,-2,2,0,-1,2,0,1,-3,0,2,-3,0,2,-2,0,3,-2,0,3,-1,0,-1,1,0,-1,2,
%W A065676 0,-2,2,0,-2,3,0,-1,3,0,-1,1,0,-3,1,0,-3,2,0,-2,2,0,-2,1,0,-1,1,0,1,-1,0,2,-1,0,2,-1,
%X A065676 0,3,-1,0,3,-4,0,1,-4,0,1,-1,0,2,-1,0,2,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-2,0,1,-2,0,1,-1
%N A065676 The exponent of 2 in the fractions of the whole ]0,inf[ Stern-Brocot tree (A007305/A047679) [1/1, 1/2, 2/1, 1/3, 2/3, 3/2, 3/1, 1/4, 2/5, 3/5, 3/4, 4/3, 5/3, 5/2, 4/1, ...].
%H A065676 <a href="http://www.research.att.com/~njas/sequences/Sindx_St.html#Stern">Index entries for sequences related to Stern's sequences</a>
%p A065676 [seq(exp_of_2(SternBrocotTreeNum(j)/SternBrocotTreeDen(j)),j=1..128)];
%Y A065676 Cf. A065675, A065674, A065658.
%Y A065676 Adjacent sequences: A065673 A065674 A065675 this_sequence A065677 A065678 A065679
%Y A065676 Sequence in context: A134663 A000925 A003985 this_sequence A003263 A135211 A029294
%K A065676 sign
%O A065676 1,8
%A A065676 Antti Karttunen Nov 22 2001

%I A003263 M0045
%S A003263 1,0,1,2,1,0,2,2,0,1,3,2,0,2,3,1,0,3,3,0,2,4,2,0,3,3,0,1,4,3,0,3,5,2,0,
%T A003263 4,4,0,2,5,3,0,3,4,1,0,4,4,0,3,6,3,0,5,5,0,2,6,4,0,4,6,2,0,5,5,0,3,6,3,
%U A003263 0,4,4,0,1,5,4,0,4,7,3,0,6,6,0,3,8,5,0,5,7,2,0,6,6,0,4,8,4,0,6,6,0,2,7
%N A003263 Number of representations of n as a sum of distinct Lucas numbers 1,3,4,7,11 ... (A000204).
%D A003263 A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 58.
%H A003263 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b003263.txt">Table of n, a(n) for n = 0..9349</a>
%Y A003263 Cf. A054770, A000204.
%Y A003263 Adjacent sequences: A003260 A003261 A003262 this_sequence A003264 A003265 A003266
%Y A003263 Sequence in context: A000925 A003985 A065676 this_sequence A135211 A029294 A065434
%K A003263 nonn,easy
%O A003263 1,4
%A A003263 njas
%E A003263 More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 29 2000

%I A135211
%S A135211 1,1,0,0,1,0,1,1,0,2,1,0,2,2,0,2,3,0,3,3,0,4,4,0,5,6,0,6,7,0,7,8,0,10,
%T A135211 10,0,13,13,0,14,16,0,17,18,0,22,22,0,26,28,0,30,33,0,36,38,0,44,45,0,
%U A135211 52,55,0,60,65,0,70,74,0,84,87,0,99,104,0,112,121,0,131,138,0,156,160
%V A135211 1,-1,0,0,-1,0,1,-1,0,2,-1,0,2,-2,0,2,-3,0,3,-3,0,4,-4,0,5,-6,0,6,-7,0,7,-8,0,10,-10,0,
%W A135211 13,-13,0,14,-16,0,17,-18,0,22,-22,0,26,-28,0,30,-33,0,36,-38,0,44,-45,0,52,-55,0,60,
%X A135211 -65,0,70,-74,0,84,-87,0,99,-104,0,112,-121,0,131,-138,0,156,-160
%N A135211 Expansion of q^(1/4) * eta(q) * eta(q^4) * eta(q^6) / ( eta(q^2) * eta(q^3) * eta(q^12) ) in powers of q.
%F A135211 Expansion of psi(-q) / psi(-q^3) in powers of q where psi() is a Ramanujan theta function.
%F A135211 Euler transform of period 12 sequence [ -1, 0, 0, -1, -1, 0, -1, -1, 0, 0, -1, 0, ...].
%F A135211 Given g.f. A(x), then B(x) = A(x^4) / x satisfies 0 = f(B(x), B(x^3)) where f(u, v) = (1 + v^4) - (1 + u*v)^3.
%F A135211 G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = sqrt(3) / f(t) where q = exp(2 pi i t).
%F A135211 a(3*n+2) = 0.
%e A135211 1/q - q^3 - q^15 + q^23 - q^27 + 2*q^35 - q^39 + 2*q^47 - 2*q^51 + ...
%o A135211 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))}
%Y A135211 -A036018(n) = a(3*n+1). Convolution inverse of A036018.
%Y A135211 Adjacent sequences: A135208 A135209 A135210 this_sequence A135212 A135213 A135214
%Y A135211 Sequence in context: A003985 A065676 A003263 this_sequence A029294 A065434 A045832
%K A135211 sign
%O A135211 0,10
%A A135211 Michael Somos, Nov 22 2007, Nov 23 2007

%I A029294
%S A029294 1,0,0,1,0,0,2,1,0,2,2,0,3,3,1,3,4,2,4,5,4,5,6,5,7,7,7,
%T A029294 9,9,8,12,11,10,14,14,12,17,17,15,19,21,18,23,24,22,26,
%U A029294 28,26,31,32,31,35,37,35,41,42,41,46,48,46,53,54,53,59
%N A029294 Expansion of 1/((1-x^3)(1-x^6)(1-x^7)(1-x^10)).
%Y A029294 Adjacent sequences: A029291 A029292 A029293 this_sequence A029295 A029296 A029297
%Y A029294 Sequence in context: A065676 A003263 A135211 this_sequence A065434 A045832 A035393
%K A029294 nonn
%O A029294 0,7
%A A029294 njas

%I A065434
%S A065434 2,1,0,2,2,0,3,9,6,3,8,7,7,1,5,5,4,9,9,2,6,2,8,4,7,9,5,9,3,8,9,6,9,
%T A065434 0,2,7,7,7,3,3,4,3,4,0,5,2,4,9,0,2,7,8,1,7,5,4,6,2,9,5,2,0,4,0,3,5,
%U A065434 8,7,5,9,8,5,8,6,0,6,8,8,9,0,7,9,9,7,1,3,6,5,8,5,1,4,1,8,0,1,5,1,4
%N A065434 Decimal expansion of imaginary part of 2nd nontrivial zero of Riemann zeta function.
%e A065434 21.0220396387715549926284795938969...
%Y A065434 See A002410 and A058303 for more information.
%Y A065434 Adjacent sequences: A065431 A065432 A065433 this_sequence A065435 A065436 A065437
%Y A065434 Sequence in context: A003263 A135211 A029294 this_sequence A045832 A035393 A068913
%K A065434 nonn,cons
%O A065434 2,1
%A A065434 njas, Nov 24 2001

%I A045832
%S A045832 1,0,2,1,0,2,2,0,6,0,0,2,2,0,8,1,0,2,4,0,8,2,0,2,5,0,6,
%T A045832 2,0,4,6,0,8,2,0,6,6,0,12,0,0,0,4,0,12,4,0,2,7,0,16,2,0,
%U A045832 6,4,0,12,0,0,8,6,0,12,1,0,4
%N A045832 A005889/3.
%Y A045832 Adjacent sequences: A045829 A045830 A045831 this_sequence A045833 A045834 A045835
%Y A045832 Sequence in context: A135211 A029294 A065434 this_sequence A035393 A068913 A128306
%K A045832 nonn
%O A045832 0,3
%A A045832 njas

%I A035393
%S A035393 0,0,0,1,0,0,1,1,0,0,2,1,0,2,2,1,0,4,2,1,3,5,2,1,7,5,2,6,9,5,2,13,10,5,
%T A035393 9,18,10,5,21,20,10,16,30,21,10,35,35,21,25,52,37,21,55,62,38,43,83,67,
%U A035393 38,88,102,69,68,135,112,70,135,168,117,112,208,188,119,209,265
%N A035393 Number of partitions of n into parts 7k or 7k+4.
%Y A035393 Adjacent sequences: A035390 A035391 A035392 this_sequence A035394 A035395 A035396
%Y A035393 Sequence in context: A029294 A065434 A045832 this_sequence A068913 A128306 A113423
%K A035393 nonn
%O A035393 1,11
%A A035393 Olivier Gerard (olivier.gerard(AT)gmail.com)

%I A068913
%S A068913 1,0,1,0,2,1,0,2,2,1,0,4,4,2,1,0,4,6,4,2,1,0,8,12,8,4,2,1,0,8,18,14,8,
%T A068913 4,2,1,0,16,36,28,16,8,4,2,1,0,16,54,48,30,16,8,4,2,1,0,32,108,96,60,
%U A068913 32,16,8,4,2,1,0,32,162,164,110,62,32,16,8,4,2,1,0,64,324,328,220,124
%N A068913 Square array read by antidiagonals of number of k step walks (each step +/-1 starting from 0) which are never more than n or less than -n.
%F A068913 Starting with T(n, 0)=1; if (k-n) is negative or even then T(n, k)=2T(n, k-1); otherwise T(n, k)=2T(n, k-1)-A061897(n-1, (k-n-1)/2); so for n>=k T(n, k)=2^k.
%e A068913 Rows start: 1,0,0,0,0,...; 1,2,2,4,4,...; 1,2,4,6,12,...; 1,2,4,8,14,... etc.
%Y A068913 Cf. early rows: A000007, A016116 (without initial term), A068911, A068912, next is effectively twice A039717 alternating with four times A039717. Central and lower diagonals are A000079, higher diagonals include A000918, A028399.
%Y A068913 Adjacent sequences: A068910 A068911 A068912 this_sequence A068914 A068915 A068916
%Y A068913 Sequence in context: A065434 A045832 A035393 this_sequence A128306 A113423 A131258
%K A068913 nonn,tabl
%O A068913 0,5
%A A068913 Henry Bottomley (se16(AT)btinternet.com), Mar 06 2002

%I A128306
%S A128306 1,1,0,2,1,0,2,2,1,0,4,6,4,1,0,2,4,6,4,1,0,6,15,20,15,6,1,0,4,12,22,24,
%T A128306 16,6,1,0,6,21,45,60,51,27,8,1,0,4,16,44,76,85,62,29,8,1,0,4,16,44,76,
%U A128306 85,62,29,8,1,0
%N A128306 Triangle, A054521 * A007318.
%C A128306 Left border = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6,...). Row sums = A054432: (1, 1, 3, 5, 15, 17, 63,...).
%F A128306 A054521 * A007318 as infinite lower triangular matrices.
%e A128306 First few rows of the triangle are:
%e A128306 1;
%e A128306 1, 0;
%e A128306 2, 1, 0;
%e A128306 2, 2, 1, 0;
%e A128306 4, 6, 4, 1, 0;
%e A128306 2, 4, 6, 4, 1, 0;
%e A128306 6, 15, 20, 15, 6, 1, 0;
%e A128306 ...
%Y A128306 Cf. A054521, A007318, A000010, A054432.
%Y A128306 Adjacent sequences: A128303 A128304 A128305 this_sequence A128307 A128308 A128309
%Y A128306 Sequence in context: A045832 A035393 A068913 this_sequence A113423 A131258 A029366
%K A128306 nonn,tabl
%O A128306 1,4
%A A128306 Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 25 2007

%I A113423
%S A113423 1,1,0,1,1,0,2,1,0,2,2,1,1,2,2,2,2,1,0,2,2,3,0,1,1,2,2,0,2,2,0,1,0,3,2,
%T A113423 2,0,1,2,4,1,1,2,4,0,2,0,0,0,0,2,3,0,3,0,2,2,1,2,2,1,0,0,7,2,0,6,2,2,2,
%U A113423 2,2,0,3,0,2,0,0,4,2,2,5,2,1,1,2,0,1,2,2,2,0,2,2,4,2,4,0,2,0,2,2,0,3,0
%V A113423 1,-1,0,-1,-1,0,2,1,0,2,-2,1,-1,2,-2,-2,-2,-1,0,2,2,-3,0,1,1,2,2,0,2,-2,0,1,0,-3,2,-2,
%W A113423 0,1,-2,-4,-1,1,2,-4,0,2,0,0,0,0,2,3,0,3,0,2,-2,-1,-2,2,-1,0,0,7,2,0,-6,-2,-2,-2,-2,-2,
%X A113423 0,-3,0,2,0,0,4,-2,-2,5,-2,-1,1,-2,0,1,2,2,-2,0,2,2,4,-2,4,0,2,0,-2,2,0,-3,0
%N A113423 Expansion of q^(-1)eta(q^2)*eta(q^8)^2*eta(q^10)/eta(q^4) in powers of q^2.
%D A113423 K. Ono, Ramanujan, taxicabs, birthdates, ZIP codes, and twists, Amer. Math. Monthly 104 (1997), no. 10, 912-917, MR1490909 (98i:11020)
%F A113423 Euler transform of period 20 sequence [ -1, 0, -1, -2, -2, 0, -1, -2, -1, -1, -1, -2, -1, 0, -2, -2, -1, 0, -1, -3, ...].
%e A113423 q -q^3 -q^7 -q^9 +2*q^13 +q^15 +2*q^19 -2*q^21 +q^23 +...
%o A113423 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^4+A)^2*eta(x^5+A)/eta(x^2+A), n))}
%Y A113423 Adjacent sequences: A113420 A113421 A113422 this_sequence A113424 A113425 A113426
%Y A113423 Sequence in context: A035393 A068913 A128306 this_sequence A131258 A029366 A112848
%K A113423 sign
%O A113423 0,7
%A A113423 Michael Somos, Oct 31 2005

%I A131258
%S A131258 1,1,1,0,2,1,0,2,2,1,1,2,3,2,1,1,3,5,3,2,1,0,4,8,6,3,2,1,0,4,11,12,6,3,
%T A131258 2,1,1,4,14,20,13,6,3,2,1,1,5,18,30,25,13,6,3,2,1
%N A131258 A129686^(-1) * A052509.
%C A131258 Row sums = A097083: (1, 2, 3, 5, 9, 15,...)
%F A131258 A129686^(-1) * A052509 as infinite lower triangular matrices, where A129686 = the alternate term operator.
%e A131258 First few rows of the triangle are:
%e A131258 1;
%e A131258 1, 1;
%e A131258 0, 2, 1;
%e A131258 0, 2, 2, 1;
%e A131258 1, 2, 3, 2, 1;
%e A131258 1, 3, 5, 3, 2, 1;
%e A131258 0, 4, 8, 6, 3, 2, 1;
%e A131258 ...
%Y A131258 Cf. A129686, A052509, A097083.
%Y A131258 Adjacent sequences: A131255 A131256 A131257 this_sequence A131259 A131260 A131261
%Y A131258 Sequence in context: A068913 A128306 A113423 this_sequence A029366 A112848 A035152
%K A131258 nonn,tabl
%O A131258 0,5
%A A131258 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 23 2007

%I A029366
%S A029366 1,0,0,0,1,0,0,1,1,0,1,2,1,0,2,2,1,1,3,2,2,3,4,2,3,4,4,
%T A029366 3,5,5,5,5,7,6,6,7,8,7,8,9,10,9,11,11,12,11,13,13,14,14,
%U A029366 16,16,17,17,19,19,20,20,22,22,24,24,26,26,28,28,30,30
%N A029366 Expansion of 1/((1-x^4)(1-x^7)(1-x^10)(1-x^11)).
%Y A029366 Adjacent sequences: A029363 A029364 A029365 this_sequence A029367 A029368 A029369
%Y A029366 Sequence in context: A128306 A113423 A131258 this_sequence A112848 A035152 A035204
%K A029366 nonn
%O A029366 0,12
%A A029366 njas

%I A112848
%S A112848 1,1,2,1,0,2,2,1,2,0,0,2,2,2,0,1,0,2,2,0,4,0,0,2,1,2,2,2,0,0,2,1,0,0,0,
%T A112848 2,2,2,4,0,0,4,2,0,0,0,0,2,3,1,0,2,0,2,0,2,4,0,0,0,2,2,4,1,0,0,2,0,0,0,
%U A112848 0,2,2,2,2,2,0,4,2,0,2,0,0,4,0,2,0,0,0,0,4,0,4,0,0,2,2,3,0,1,0,0,2,2,0
%V A112848 1,-1,-2,1,0,2,2,-1,-2,0,0,-2,2,-2,0,1,0,2,2,0,-4,0,0,2,1,-2,-2,2,0,0,2,-1,0,0,0,-2,2,
%W A112848 -2,-4,0,0,4,2,0,0,0,0,-2,3,-1,0,2,0,2,0,-2,-4,0,0,0,2,-2,-4,1,0,0,2,0,0,0,0,2,2,-2,-2,
%X A112848 2,0,4,2,0,-2,0,0,-4,0,-2,0,0,0,0,4,0,-4,0,0,2,2,-3,0,1,0,0,2,-2,0
%N A112848 Expansion of eta(q)*eta(q^2)*eta(q^18)^2/(eta(q^6)*eta(q^9)) in powers of q.
%F A112848 Euler transform of period 18 sequence [ -1, -2, -1, -2, -1, -1, -1, -2, 0, -2, -1, -1, -1, -2, -1, -2, -1, -2, ...].
%F A112848 Moebius transform is period 18 sequence [1, -2, -3, 2, -1, 6, 1, -2, 0, 2, -1, -6, 1, -2, 3, 2, -1, 0, ...].
%F A112848 Multiplicative with a(2^e) = (-1)^e, a(3^e) = -2 if e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
%F A112848 G.f.: Sum_{k>0} kronecker(-3, k) x^k(1-x^(2k))^2/(1-x^(6k)) = x Product_{k>0} (1-x^k)(1-x^(2k))(1+x^(9k))(1+x^(6k)+x^(12k)).
%F A112848 a(3n)=-2*A092829(n). a(3n+1)=A093829(3n+1)=A033687(n). a(3n+2)=A093829(3n+2). a(6n)/2=A093829(n). a(6n+1)=A097195(n). a(6n+3)=-2*A033762(n). a(6n+5)=0.
%o A112848 (PARI) {a(n)=if(n<1, 0, if(n%3==0, n/=3; -2,1)* sumdiv(n,d,kronecker(-12,d) -if(d%2==0, 2*kronecker(-3,d/2))))}
%o A112848 (PARI) {a(n)=local(A); if (n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)*eta(x^2+A)*eta(x^18+A)^2/ eta(x^6+A)/eta(x^9+A), n))}
%Y A112848 Adjacent sequences: A112845 A112846 A112847 this_sequence A112849 A112850 A112851
%Y A112848 Sequence in context: A113423 A131258 A029366 this_sequence A035152 A035204 A016154
%K A112848 sign,mult
%O A112848 1,3
%A A112848 Michael Somos, Sep 22 2005

%I A035152
%S A035152 1,1,2,1,0,2,2,1,3,0,0,2,2,2,0,1,2,3,1,0,4,0,2,2,1,2,4,
%T A035152 2,2,0,0,1,0,2,0,3,2,1,4,0,0,4,0,0,0,2,2,2,3,1,4,2,2,4,
%U A035152 0,2,2,2,2,0,0,0,6,1,0,0,2,2,4,0,0,3,2,2,2,1,0,4,0,0,5
%N A035152 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -38.
%o A035152 (PARI) direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
%Y A035152 Adjacent sequences: A035149 A035150 A035151 this_sequence A035153 A035154 A035155
%Y A035152 Sequence in context: A131258 A029366 A112848 this_sequence A035204 A016154 A029343
%K A035152 nonn
%O A035152 1,3
%A A035152 njas

%I A035204
%S A035204 1,1,2,1,0,2,2,1,3,0,1,2,2,2,0,1,0,3,0,0,4,1,0,2,1,2,4,
%T A035204 2,2,0,0,1,2,0,0,3,0,0,4,0,0,4,0,1,0,0,0,2,3,1,0,2,0,4,
%U A035204 0,2,0,2,2,0,2,0,6,1,0,2,2,0,0,0,0,3,0,0,2,0,2,4,2,0,5
%N A035204 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 22.
%o A035204 (PARI) direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
%Y A035204 Adjacent sequences: A035201 A035202 A035203 this_sequence A035205 A035206 A035207
%Y A035204 Sequence in context: A029366 A112848 A035152 this_sequence A016154 A029343 A137992
%K A035204 nonn
%O A035204 1,3
%A A035204 njas

%I A016154
%S A016154 1,1,1,0,0,1,1,1,0,0,0,1,1,2,1,0,2,2,2,1,0,1,1,0,1,1,0,
%T A016154 1,2,2,1,1,2,3,2,1,1,1,1,1,1,1,1,2,4,3,2,1,2,3,1,0,2,1,
%U A016154 0,3,3,3,0,2,4,3,1,2,3,3,1,1,3,2,0,2,3,2,0,3,4,4,1,1,3
%V A016154 1,1,1,0,0,-1,-1,-1,0,0,0,-1,-1,-2,-1,0,2,2,2,1,0,-1,-1,0,1,1,0,
%W A016154 -1,-2,-2,-1,1,2,3,2,1,-1,-1,-1,1,1,1,-1,-2,-4,-3,-2,1,2,3,1,0,-2,-1,
%X A016154 0,3,3,3,0,-2,-4,-3,-1,2,3,3,1,-1,-3,-2,0,2,3,2,0,-3,-4,-4,-1,1,3
%N A016154 Inverse of 2145th cyclotomic polynomial.
%p A016154 with(numtheory,cyclotomic); c := n->series(1/cyclotomic(n,x),x,80);
%Y A016154 Adjacent sequences: A016151 A016152 A016153 this_sequence A016155 A016156 A016157
%Y A016154 Sequence in context: A112848 A035152 A035204 this_sequence A029343 A137992 A047654
%K A016154 sign
%O A016154 0,14
%A A016154 Simon Plouffe (plouffe(AT)math.uqam.ca)

%I A029343
%S A029343 1,0,0,0,1,1,0,0,1,2,1,0,2,2,2,1,2,3,3,2,3,4,4,3,5,5,5,
%T A029343 5,6,7,7,6,8,9,9,8,11,11,11,11,13,14,14,13,16,18,17,16,
%U A029343 20,21,21,20,23,25,26,24,27,30,30,29,33,34,35,35,38,40
%N A029343 Expansion of 1/((1-x^4)(1-x^5)(1-x^9)(1-x^12)).
%Y A029343 Adjacent sequences: A029340 A029341 A029342 this_sequence A029344 A029345 A029346
%Y A029343 Sequence in context: A035152 A035204 A016154 this_sequence A137992 A047654 A058487
%K A029343 nonn
%O A029343 0,10
%A A029343 njas

%I A137992
%S A137992 1,2,1,0,2,2,2,2,1,0,2,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,1,0,2,0,1,2,2,2,2,
%T A137992 0,1,2,1,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%U A137992 2,2,2,2,2,2,2,2,2,2
%N A137992 A014137 (= partial sums of Catalan numbers A000108) mod 3.
%C A137992 As usual, "mod 3" means to chose the unique representative in { 0,1,2 } of the equivalence class modulo 3Z.
%F A137992 a(n) = sum( k=0..n, C(k) ) (mod 3), where C(k) = binomial(2k,k)/(k+1)
%F A137992 a(n) = 1 <=> n = 2 A137821(m) for some m (with A137821(0):=0).
%o A137992 (PARI) A137992(n) = lift( sum( k=0,n, binomial( 2*k,k )/(k+1), Mod(0,3) ))
%Y A137992 Cf. A014137, A000108, A137821-A137824, A107755; A014138(n)+1 = a(n+1) (mod 3).
%Y A137992 Adjacent sequences: A137989 A137990 A137991 this_sequence A137993 A137994 A137995
%Y A137992 Sequence in context: A035204 A016154 A029343 this_sequence A047654 A058487 A062243
%K A137992 easy,nonn
%O A137992 0,2
%A A137992 M. F. Hasler (MHasler(AT)univ-ag.fr), Mar 16 2008

%I A047654
%S A047654 1,2,1,0,2,2,2,2,1,0,2,2,3,0,2,0,0,2,2,0,2,2,1,0,0,2,2,2,1,2,0,
%T A047654 2,2,0,2,0,2,0,2,0,0,0,1,2,0,0,2,0,2,0,1,2,0,2,2,2,0,2,0,2,0,2,
%U A047654 2,0,4,0,0,2,1,2,0,2,0,0,0,0,2,4,1,0,0,2,2,2,2,0,0,2,0,2,2,2,2
%V A047654 1,-2,1,0,-2,2,-2,2,1,0,2,-2,3,0,2,0,0,2,-2,0,-2,2,-1,0,0,-2,-2,-2,1,-2,0,
%W A047654 -2,-2,0,2,0,-2,0,-2,0,0,0,1,2,0,0,2,0,2,0,1,2,0,-2,2,2,0,2,0,2,0,2,
%X A047654 2,0,-4,0,0,2,1,-2,0,-2,0,0,0,0,2,-4,1,0,0,-2,-2,-2,-2,0,0,-2,0,2,-2,2,-2
%N A047654 Expand {Product_{j=1..inf} (1-x^j) - 1 }^2 in powers of x.
%D A047654 H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
%Y A047654 Adjacent sequences: A047651 A047652 A047653 this_sequence A047655 A047656 A047657
%Y A047654 Sequence in context: A016154 A029343 A137992 this_sequence A058487 A062243 A128095
%K A047654 sign
%O A047654 1,2
%A A047654 njas

%I A058487
%S A058487 1,2,1,0,2,2,2,4,3,4,8,4,5,14,7,8,20,12,14,28,17,20,44,24,28,66,36,40,
%T A058487 90,52,56,124,71,80,176,96,109,244,133,144,326,182,198,432,240,268,580,
%U A058487 316,349,772,420,456,1004,552,600,1300,713,780,1692,916,1001,2186,1182
%V A058487 1,2,1,0,-2,-2,2,4,3,-4,-8,-4,5,14,7,-8,-20,-12,14,28,17,-20,-44,-24,28,66,36,-40,-90,
%W A058487 -52,56,124,71,-80,-176,-96,109,244,133,-144,-326,-182,198,432,240,-268,-580,-316,349,
%X A058487 772,420,-456,-1004,-552,600,1300,713,-780,-1692,-916,1001,2186,1182
%N A058487 McKay-Thompson series of class 12I for Monster.
%C A058487 Euler transform of period 6 sequence [2,-2,0,-2,2,0,...]. - Michael Somos Mar 18 2004
%C A058487 G.f. A(x) satisfies 0=f(A(x^2)/x,A(x^4)/x^2) where f(u,v)=u^2+3v-u^2v+v^2. - Michael Somos Mar 18 2004
%C A058487 Expansion of q(eta(q^4)^4*eta(q^6)^2/(eta(q^2)^2*eta(q^12)^4)) in powers of q^2.
%D A058487 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%D A058487 J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.
%F A058487 G.f.: ( Product_{k>0} (1-x^(6k-2))(1-x^(6k-4))/((1-x^(6k-1))(1-x^(6k-5))) )^2.
%e A058487 eta(4z)^4*eta(6z)^2/(eta(2z)^2*eta(12z)^4) = 1/q + 2*q^1 + 1*q^3 - 2*q^7 - 2*q^9 + 2*q^11 + 4*q^13 + 3*q^15 + ...
%e A058487 T12I = 1/q + 2*q + 1*q^3 - 2*q^7 - 2*q^9 + 2*q^11 + 4*q^13 + 3*q^15 + ...
%o A058487 (PARI) a(n)=local(A,m); if(n<-1,0,n++; A=1+O(x); m=1; while(m<=n,m*=2; A=subst(A,x,x^2); A=sqrt(A*(A+3*x)/(A-x))); polcoeff(A,n))
%o A058487 (PARI) a(n)=local(A); if(n<0,0,A=x*O(x^n); polcoeff((eta(x^2+A)^2*eta(x^3+A)/eta(x+A)/eta(x^6+A)^2)^2,n))
%Y A058487 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%Y A058487 A062243(n)=(-1)^n*a(n).
%Y A058487 Adjacent sequences: A058484 A058485 A058486 this_sequence A058488 A058489 A058490
%Y A058487 Sequence in context: A029343 A137992 A047654 this_sequence A062243 A128095 A097854
%K A058487 sign
%O A058487 0,2
%A A058487 njas, Nov 27 2000

%I A062243
%S A062243 1,2,1,0,2,2,2,4,3,4,8,4,5,14,7,8,20,12,14,28,17,20,44,24,28,66,36,40,
%T A062243 90,52,56,124,71,80,176,96,109,244,133,144,326,182,198,432,240,268,580,
%U A062243 316,349,772,420,456,1004,552,600,1300,713,780,1692,916,1001,2186,1182
%V A062243 1,-2,1,0,-2,2,2,-4,3,4,-8,4,5,-14,7,8,-20,12,14,-28,17,20,-44,24,28,-66,36,40,-90,52,
%W A062243 56,-124,71,80,-176,96,109,-244,133,144,-326,182,198,-432,240,268,-580,316,349,-772,
%X A062243 420,456,-1004,552,600,-1300,713,780,-1692,916,1001,-2186,1182
%N A062243 McKay-Thompson series of class 24c for the Monster group.
%C A062243 Euler transform of period 12 sequence [ -2,0,0,-2,-2,0,-2,-2,0,0,-2,0,...]. - Michael Somos May 14 2004
%C A062243 Expansion of Hauptmodul for Gamma'_0(12).
%D A062243 J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.
%F A062243 G.f.: ( Product_{k>0} (1-x^(4k))(1-x^(2k-1))/(1-x^(3k)) )^2.
%F A062243 Given G.f. A(x), then B(x)=A(x^2)^2/(3x^2) satisfies 0=f(B(x), B(x^2)) where f(u, v)= (u+v)^2(u^2+v^2-uv) +3(u^3+v^3)(1+uv) -9uv(1+(uv)^2) -90(uv)^2 -27uv(u+v)(1+uv). - Michael Somos May 14 2004
%F A062243 Expansion of q^(1/2)(eta(q)eta(q^4)eta(q^6)/(eta(q^2)eta(q^3)eta(q^12)))^2 in powers of q.
%e A062243 T24c = 1/q -2*q +q^3 -2*q^7 +2*q^9 +2*q^11 -4*q^13 +3*q^15 +...
%o A062243 (PARI) a(n)=local(A); if(n<0,0,A=x*O(x^n); polcoeff((eta(x+A)*eta(x^4+A)*eta(x^6+A)/eta(x^2+A)/eta(x^3+A)/eta(x^12+A))^2,n))
%Y A062243 A058487(n)=(-1)^n*a(n).
%Y A062243 Adjacent sequences: A062240 A062241 A062242 this_sequence A062244 A062245 A062246
%Y A062243 Sequence in context: A137992 A047654 A058487 this_sequence A128095 A097854 A144027
%K A062243 sign
%O A062243 0,2
%A A062243 njas, Jul 01 2001

%I A128095
%S A128095 1,0,1,0,1,1,0,1,2,1,0,2,2,3,1,0,4,4,4,4,1,0,8,8,8,7,5,1,0,17,16,17,14,
%T A128095 11,6,1,0,37,34,36,31,23,16,7,1,0,82,74,79,68,53,36,22,8,1,0,185,164,
%U A128095 177,152,121,86,54,29,9,1,0,423,370,402,346,278,204,134,78,37,10,1,0
%N A128095 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n and having k steps that touch the x-axis (1<=k<=n).
%C A128095 T(n,k)=number of secondary structures of size n in which the shortest path from one end to the other one has length k-1. Row sums yield A004148. T(n,2)=A004148(n-2). T(n,3)=2*A004148(n-3) for n>=4. Sum(k*T(n,k),k=1..n)=A128096(n).
%F A128095 G.f.=2/[2-2tz-t^2+t^2*z+t^2*z^2+t^2*sqrt((1+z+z^2)(1-3z+z^2))]-1.
%e A128095 T(5,4)=3 because we have HU(H)DH, HHU(H)D, and U(H)DHH, where U=(1,1), H=(1,0), and D=(1,-1) and the steps that do not touch the x-axis are shown between parentheses.
%e A128095 Triangle starts:
%e A128095 1;
%e A128095 0,1;
%e A128095 0,1,1;
%e A128095 0,1,2,1;
%e A128095 0,2,2,3,1;
%e A128095 0,4,4,4,4,1;
%e A128095 0,8,8,8,7,5,1;
%p A128095 G:=2/(2-2*t*z-t^2+t^2*z+t^2*z^2+t^2*sqrt((1+z+z^2)*(1-3*z+z^2)))-1: Gser:=simplify(series(G,z=0,15)): for n from 1 to 13 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 1 to 13 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form
%Y A128095 Cf. A004148, A128096.
%Y A128095 Adjacent sequences: A128092 A128093 A128094 this_sequence A128096 A128097 A128098
%Y A128095 Sequence in context: A047654 A058487 A062243 this_sequence A097854 A144027 A019591
%K A128095 nonn,tabl
%O A128095 1,9
%A A128095 Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 14 2007

%I A097854
%S A097854 1,1,0,1,0,1,1,0,1,2,1,0,2,2,4,1,0,4,4,4,8,1,0,9,8,8,8,17,1,0,21,18,16,
%T A097854 16,17,38,1,0,51,42,36,32,34,38,89,1,0,127,102,84,72,68,76,89,216,1,0,
%U A097854 323,254,204,168,153,152,178,216,539,1,0,835,646,508,408,357,342,356
%N A097854 Triangle read by rows: T(n,k)=number of Motzkin paths of length n and having abscissa of first return (i.e. first down step hitting the x-axis) equal to k (k>0); T(n,0)=1 (accounts for the paths consisting only of level steps).
%C A097854 Row sums are the Motzkin numbers (A001006).
%F A097854 G.f.=(1-tz+t^2*z^2*M(tz)M(z)-t^2*z^3*M(tz)M(z))/(1-z-tz+tz^2), where M(z)=(1-z-sqrt(1-2z-3z^2))/(2z^2) is the g.f. of the Motzkin numbers. T(n, k) = m[n-k]*sum(m[j], j=0..k-2), where m[n]=A001006(n) are the Motzkin numbers.
%e A097854 Triangle starts:
%e A097854 1;
%e A097854 1,0;
%e A097854 1,0,1;
%e A097854 1,0,1,2;
%e A097854 1,0,2,2,4;
%e A097854 1,0,4,4,4,8;
%e A097854 Row n has n+1 terms.
%e A097854 T(5,3)=4 because the Motzkin paths of length 5 and having abscissa of first return equal to 3 are HU(D)HH, HU(D)UD, UH(D)HH, and UH(D)UD (first returns to axis shown between parentheses); here U=(1,1), H=(1,0) and D=(1,-1).
%p A097854 G:=(1-t*z+t^2*z^2*M(t*z)*M(z)-t^2*z^3*M(t*z)*M(z))/(1-z-t*z+t*z^2): M:=z->(1-z-sqrt(1-2*z-3*z^2))/2/z^2: Gser:=simplify(series(G,z=0,14)): P[0]:=1: for n from 1 to 13 do P[n]:=coeff(Gser,z^n) od: seq(seq(coeff(t*P[n],t^k),k=1..n+1),n=0..12); M:=(1-z-sqrt(1-2*z-3*z^2))/2/z^2: Mser:=series(M,z=0,15): m[0]:=1: for n from 1 to 12 do m[n]:=coeff(Mser,z^n) od: T:=proc(n,k) if k=0 then 1 elif k<=n then m[n-k]*sum(m[j],j=0..k-2) else 0 fi end: TT:=(n,k)->T(n-1,k-1): matrix(11,11,TT); # generates the triangle:
%Y A097854 Cf. A001006.
%Y A097854 Adjacent sequences: A097851 A097852 A097853 this_sequence A097855 A097856 A097857
%Y A097854 Sequence in context: A058487 A062243 A128095 this_sequence A144027 A019591 A091967
%K A097854 nonn,tabf
%O A097854 0,10
%A A097854 Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 31 2004

%I A144027
%S A144027 1,1,1,0,1,2,1,0,2,3,0,1,0,3,6,0,0,2,0,6,10,1,0,0,3,0,10,18,1,1,0,0,6,0,
%T A144027 18,32,0,1,2,0,0,10,0,32,58,0,0,2,3,0,0,18,0,58,103,1,0,0,3,6,0,0,32,0,
%U A144027 103,184,0,1,0,0,6,10,0,0,58,0,184,329,1,0,2,0,0,10,18,0,0,103,329,588
%N A144027 Eigentriangle by rows, T(n,k) = A010060(n-k+1)*A144026(k-1), 1<=k<=n.
%C A144027 Left column = the Thue-Morse sequence A010060 starting with offset 1.
%C A144027 Right border = A144026 shifted: (1, 1, 2, 3, 6, 10, 18,...).
%C A144027 Row sums = A144026: (1, 2, 3, 6, 10, 18,...).
%C A144027 Sum of n-th row terms = rightmost term of next row.
%F A144027 Eigentriangle by rows, T(n,k) = A010060(n-k+1)*A144026(k-1), 1<=k<=n.
%F A144027 The triangle is generated from the Thue-Morse sequence A010060 using offset 1:
%F A144027 (1, 1, 0, 1, 0, 0, 1,...). A144026 is shifted to (1, 1, 2, 3, 6, 10, 18,...).
%e A144027 The first few rows of the triangle =
%e A144027 1;
%e A144027 1, 1;
%e A144027 0, 1, 2;
%e A144027 1, 0, 2, 3;
%e A144027 0, 1, 0, 3, 6;
%e A144027 0, 0, 2, 0, 6, 10;
%e A144027 1, 0, 0, 3, 0, 10, 18;
%e A144027 1, 1, 0, 0, 6, 0, 18, 32;
%e A144027 0, 1, 2, 0, 0, 10, 0, 32, 58;
%e A144027 0, 0, 2, 3, 0, 0, 18, 0, 58, 103;
%e A144027 1, 0, 0, 3, 6, 0, 0, 32, 0, 103, 184;
%e A144027 ...
%e A144027 Row 4 = (1, 0, 2, 3) = termwise products of (1, 0, 1, 1) and (1, 1, 2, 3), where (1, 0, 1, 1) = the first 4 terms of A010060, reversed with offset 1.
%e A144027 (1, 1, 2, 3) = first 4 terms of A144026 shifted: (1, 1, 2, 3, 6, 10, 18,...).
%Y A144027 A010060, Cf. 144026
%Y A144027 Sequence in context: A062243 A128095 A097854 this_sequence A019591 A091967 A031135
%K A144027 nonn,tabl
%O A144027 1,6
%A A144027 Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 07 2008

%I A019591
%S A019591 0,0,1,0,2,1,0,2,3,0,1,4,0,2,4,0,2,3,0,1,4,0,2,4,0,2,6,0,1,3,0,2,5,
%T A019591 0,2,6,0,1,8,0,2,3,0,2,6,0,1,4,0,2,4,0,2,3,0,1,4,0,2,8,0,2,8,0,1,3,
%U A019591 0,2,5,0,2,7,0,1,8,0,2,3,0,2,8,0,1,4,0,2,4,0,2,3,0,1,4,0,2,8,0,2,6
%N A019591 Grundy function of game in which each player has to divide precisely one set of coins into two subsets of different sizes.
%Y A019591 Adjacent sequences: A019588 A019589 A019590 this_sequence A019592 A019593 A019594
%Y A019591 Sequence in context: A128095 A097854 A144027 this_sequence A091967 A031135 A037181
%K A019591 nonn
%O A019591 1,5
%A A019591 micheluc(AT)maia.emse.fr (Dominique MICHELUCCI)

%I A091967
%S A091967 1,2,1,0,2,3,0,6,6,4,44,1,180,42,16,1096,7652,13781,8,24000,119779,
%T A091967 458561,152116956851941670912,1054535,53,10,27,59,4806078,2,35792568,
%U A091967 3010349,2387010102192469724605148123694256128,2,0,53,43,0,4097,173,37338,42
%V A091967 1,2,1,0,2,3,0,6,6,4,44,1,180,42,16,1096,7652,13781,8,24000,119779,
%W A091967 458561,152116956851941670912,1054535,-53,10,27,59,4806078,2,35792568,
%X A091967 3010349,2387010102192469724605148123694256128,2,0,53,43,0,-4097,173,37338,42
%N A091967 a(n) = n-th term of sequence A_n.
%C A091967 This version ignores the offset of A_n, and just counts from the beginning of the terms shown in the OEIS entry.
%C A091967 Thus a(8) = 6 because A_8 begins 1,1,2,2,3,4,5,6,... [even though A_8(8) is really 7].
%H A091967 E. M. Rains and N. J. A. Sloane, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On Cayley's Enumeration of Alkanes (or 4-Valent Trees).</a>, J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
%H A091967 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H A091967 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/">Online Encyclopedia of Integer Sequences</a>
%e A091967 a(26) = 26 because the n-th term of A000026 = 26
%Y A091967 See A051070, A107357, A102288 for other versions.
%Y A091967 Cf. A000001, A000002, A000003, A000004, A000005, A000006, A000007, A000008, A000009, A000010, A000011, A000012, A000013, A000014, A000015, etc.
%Y A091967 Adjacent sequences: A091964 A091965 A091966 this_sequence A091968 A091969 A091970
%Y A091967 Sequence in context: A097854 A144027 A019591 this_sequence A031135 A037181 A051070
%K A091967 sign
%O A091967 1,2
%A A091967 Proposed by several people, including Jeff Burch (jmburch(AT)osprey.smcm.edu) and Michael Joseph Halm (hierogamous(AT)lycos.com)
%E A091967 Corrected and extended by Jud McCranie (j.mccranie(AT)comcast.net). Further extended by njas and E. M. Rains Dec 08 1998.
%E A091967 Corrected and extended by njas, May 25, 2005
%E A091967 a(43) is presently unknown, since A000043(43) is the exponent of the 43rd Mersenne prime. a(44) = 413927966. - njas, May 25 2005

%I A031135
%S A031135 1,2,1,0,2,3,0,6,8,4,63,1,316,42,16,2048,7652,13781,8,24000,11977,
%T A031135 458561,152116956851941670912,1054535,53,26,27,59,4806078,3,35792568,3010349,
%U A031135 2387010102192469724605148123694256128,2,0,53,43,0
%N A031135 Incorrect version of A091967.
%C A031135 Probably because of changes in the offsets of certain sequences, this is now incorrect. See A051070, A091967, A107357, A102288 for better versions.
%Y A031135 Adjacent sequences: A031132 A031133 A031134 this_sequence A031136 A031137 A031138
%Y A031135 Sequence in context: A144027 A019591 A091967 this_sequence A037181 A051070 A104041
%K A031135 dead
%O A031135 1,2

%I A037181
%S A037181 1,2,1,0,2,3,0,6,8,4,63,1,316,42,16,2048,7652,13781,8,24000,11977,
%T A037181 458561,152116956851941670912,1054535,53,26,27,59,4806078,3,35792568,
%U A037181 3010349,2387010102192469724605148123694256128,2,0,53,43,0,4696,173,44583,1111111111111111111111111111111111111111111
%N A037181 Incorrect version of A107357.
%Y A037181 Adjacent sequences: A037178 A037179 A037180 this_sequence A037182 A037183 A037184
%Y A037181 Sequence in context: A019591 A091967 A031135 this_sequence A051070 A104041 A104402
%K A037181 dead
%O A037181 1,2

%I A051070
%S A051070 1,2,1,0,2,3,0,7,8,4,63,1,316,78,16,2048,7652,26627,8,24000,232919,
%T A051070 1145406,3498690007594650042368,2058537,58,26,27,59,9272780,3,69273668,
%U A051070 4870847,2387010102192469724605148123694256128,1,1,53,43,0,4696,173,44583,42
%V A051070 1,2,1,0,2,3,0,7,8,4,63,1,316,78,16,2048,7652,26627,8,24000,232919,
%W A051070 1145406,3498690007594650042368,2058537,58,26,27,59,9272780,3,69273668,
%X A051070 4870847,2387010102192469724605148123694256128,1,1,53,43,0,-4696,173,44583,42
%N A051070 a(n) is the n-th term in sequence A_n, respecting the offset.
%H A051070 E. M. Rains and N. J. A. Sloane, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On Cayley's Enumeration of Alkanes (or 4-Valent Trees).</a>, J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
%H A051070 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H A051070 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/">Online Encyclopedia of Integer Sequences</a>
%Y A051070 Cf. A000001 through A000044. See also A031214, A037181, A031135.
%Y A051070 Adjacent sequences: A051067 A051068 A051069 this_sequence A051071 A051072 A051073
%Y A051070 Sequence in context: A091967 A031135 A037181 this_sequence A104041 A104402 A131084
%K A051070 easy,sign
%O A051070 1,2
%A A051070 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 23 2000
%E A051070 Rechecked and 4 more terms added by njas, May 25 2005
%E A051070 a(43) is presently unknown, since A000043(43) is the exponent of the 43rd Mersenne prime. a(44) = 668803781. - njas, May 25 2005

%I A104041
%S A104041 1,1,1,0,2,1,0,2,3,1,0,0,4,4,1,0,0,4,8,5,1,0,0,0,8,12,6,1,0,0,0,8,20,18,7,1,
%T A104041 0,0,0,0,16,32,24,8,1,0,0,0,0,16,48,56,32,9,1,0,0,0,0,0,32,80,80,40,10,1,0,0,0,
%U A104041 0,0,32,112,160,120,50,11,1,0,0,0,0,0,0,64,192,240,160,60,12,1
%V A104041 1,-1,1,0,-2,1,0,2,-3,1,0,0,4,-4,1,0,0,-4,8,-5,1,0,0,0,-8,12,-6,1,0,0,0,8,-20,18,-7,1,
%W A104041 0,0,0,0,16,-32,24,-8,1,0,0,0,0,-16,48,-56,32,-9,1,0,0,0,0,0,-32,80,-80,40,-10,1,0,0,0,
%X A104041 0,0,32,-112,160,-120,50,-11,1,0,0,0,0,0,0,64,-192,240,-160,60,-12,1
%N A104041 Triangular matrix T, read by rows, such that column k is equal (in absolute value) to row (k-1) in the matrix inverse T^-1 (with extrapolated zeros) for k>0, with T(n,n)=1 (n>=0) and T(n,n-1)=-n (n>=1).
%C A104041 Row sums are: {1,0,-1,0, 1,0,-1,0, ...}. Absolute row sums form A038754. Let A(x,y) be the g.f. of T and B(x,y) be the g.f. of T^-1; then B(x,y)=1+x*y*A(-1/y,-x*y^2) and A(x,y)=(B(-x^2*y,-1/x)-1)/(x*y).
%F A104041 G.f.: A(x, y) = (1-x+x*y)/(1+2*x^2*y-x^2*y^2).
%e A104041 Rows of T begin:
%e A104041 1;
%e A104041 -1,1;
%e A104041 0,-2,1;
%e A104041 0,2,-3,1;
%e A104041 0,0,4,-4,1;
%e A104041 0,0,-4,8,-5,1;
%e A104041 0,0,0,-8,12,-6,1;
%e A104041 0,0,0,8,-20,18,-7,1; ...
%e A104041 The matrix inverse T^-1 equals triangle A104040:
%e A104041 1;
%e A104041 1,1;
%e A104041 2,2,1;
%e A104041 4,4,3,1;
%e A104041 8,8,8,4,1;
%e A104041 16,16,20,12,5,1;
%e A104041 32,32,48,32,18,6,1;
%e A104041 64,64,112,80,56,24,7,1; ...
%e A104041 the rows of T^-1 equal columns of T in absolute value.
%o A104041 (PARI) {T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff((1-X+X*Y)/(1+2*X^2*Y-X^2*Y^2),n,x),k,y)}
%Y A104041 Cf. A104040, A038754.
%Y A104041 Adjacent sequences: A104038 A104039 A104040 this_sequence A104042 A104043 A104044
%Y A104041 Sequence in context: A031135 A037181 A051070 this_sequence A104402 A131084 A143067
%K A104041 sign,tabl
%O A104041 0,5
%A A104041 Paul D. Hanna (pauldhanna(AT)juno.com), Mar 02 2005

%I A104402
%S A104402 1,1,1,1,2,1,0,2,3,1,0,1,4,4,1,0,0,3,7,5,1,0,0,1,7,11,6,1,0,0,0,4,14,16,
%T A104402 7,1,0,0,0,1,11,25,22,8,1,0,0,0,0,5,25,41,29,9,1,0,0,0,0,1,16,50,63,37,
%U A104402 10,1,0,0,0,0,0,6,41,91,92,46,11,1,0,0,0,0,0,1,22,91,154,129,56,12,1
%V A104402 1,-1,1,1,-2,1,0,2,-3,1,0,-1,4,-4,1,0,0,-3,7,-5,1,0,0,1,-7,11,-6,1,0,0,0,4,-14,16,-7,1,
%W A104402 0,0,0,-1,11,-25,22,-8,1,0,0,0,0,-5,25,-41,29,-9,1,0,0,0,0,1,-16,50,-63,37,-10,1,0,0,0,
%X A104402 0,0,6,-41,91,-92,46,-11,1,0,0,0,0,0,-1,22,-91,154,-129,56,-12,1
%N A104402 Matrix inverse of triangle A091491, read by rows.
%C A104402 Row sums are all zeros for n>0. Absolute row sums form 2*A000045(n+1) for n>0, where A000045 = Fibonacci numbers. Sums of squared terms in row n = 2*A003440(n) for n>0, where A003440 = number of binary vectors with restricted repetitions.
%F A104402 G.f.: (1-x+x^2)/(1-x*y*(1-x)). T(n, k) = T(n-1, k-1) - T(n-2, k-1) for k>0 with T(0, 0)=1, T(1, 0)=-1, T(2, 0)=1, T(n, 0)=0 (n>2).
%F A104402 T(n, k) = (-1)^(n-k)*(C(k, n-k) + C(k+1, n-k-1)) for n>0, with T(0, 0)=1.
%e A104402 Rows begin:
%e A104402 1;
%e A104402 -1,1;
%e A104402 1,-2,1;
%e A104402 0,2,-3,1;
%e A104402 0,-1,4,-4,1;
%e A104402 0,0,-3,7,-5,1;
%e A104402 0,0,1,-7,11,-6,1;
%e A104402 0,0,0,4,-14,16,-7,1;
%e A104402 0,0,0,-1,11,-25,22,-8,1; ...
%o A104402 (PARI) {T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff((1-X+X^2)/(1-X*Y*(1-X)),n,x),k,y)} (PARI) {T(n,k)=if(n<k|k<0,0,if(n==k,1,if(n==1&k==0,-1,if(n==2&k==0,1, T(n-1,k-1)-T(n-2,k-1)))))}
%o A104402 (PARI) T(n,k)=(-1)^(n-k)*(binomial(k,n-k)+binomial(k+1,n-k-1))
%Y A104402 Cf. A091491, A000045, A003440.
%Y A104402 Adjacent sequences: A104399 A104400 A104401 this_sequence A104403 A104404 A104405
%Y A104402 Sequence in context: A037181 A051070 A104041 this_sequence A131084 A143067 A123949
%K A104402 sign,tabl
%O A104402 0,5
%A A104402 Paul D. Hanna (pauldhanna(AT)juno.com), Mar 05 2005

%I A131084
%S A131084 1,1,1,0,2,1,0,2,3,1,0,2,5,4,1,0,2,7,9,5,1,0,2,9,16,14,6,1,0,2,11,25,30,
%T A131084 20,7,1,0,2,13,36,55,50,27,8,1,0,2,15,49,91,105,77,35,9,1
%N A131084 A129686 * A007318.
%C A131084 Row sums = A098011 starting (1, 2, 3, 6, 12, 24, 48,...). A131085 = A007318 * A129686
%F A131084 A129686(signed): (1,1,1,...) in the main diagonal and (-1,-1,-1,...) in the subsubdiagonal); * A007318, Pascal's triangle; as infinite lower triangular matrices.
%e A131084 First few rows of the triangle are:
%e A131084 1;
%e A131084 1, 1;
%e A131084 0, 2, 1;
%e A131084 0, 2, 3, 1;
%e A131084 0, 2, 5, 4, 1;
%e A131084 0, 2, 7, 9, 5, 1;
%e A131084 0, 2, 9, 16, 14, 6, 1;
%e A131084 0, 2, 11, 25, 30, 20, 7, 1;
%e A131084 ...
%Y A131084 Cf. A129686, A098011, A131085.
%Y A131084 Adjacent sequences: A131081 A131082 A131083 this_sequence A131085 A131086 A131087
%Y A131084 Sequence in context: A051070 A104041 A104402 this_sequence A143067 A123949 A004718
%K A131084 nonn,tabl
%O A131084 1,5
%A A131084 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 14 2007

%I A143067
%S A143067 1,0,0,1,1,0,0,1,2,1,0,2,3,1,0,3,5,2,1,5,7,3,1,7,11,5,2,11,15,7,4,15,22,
%T A143067 11,6,22,30,15,9,30,42,22,14,42,56,31,20,56,77,43,29,77,101,58,41,101,
%U A143067 135,80,57,135,176,106,78,177,231,142,107,232,297,187,143,299,385,246
%V A143067 1,0,0,-1,1,0,0,-1,2,-1,0,-2,3,-1,0,-3,5,-2,1,-5,7,-3,1,-7,11,-5,2,-11,15,-7,4,-15,22,
%W A143067 -11,6,-22,30,-15,9,-30,42,-22,14,-42,56,-31,20,-56,77,-43,29,-77,101,-58,41,-101,135,
%X A143067 -80,57,-135,176,-106,78,-177,231,-142,107,-232,297,-187,143,-299,385,-246
%N A143067 Expansion of q^(-5/24) * eta(q^3) * eta(q^12) / (eta(q^4) * eta(q^6)) in powers of q.
%D A143067 S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41
%F A143067 Expansion of f(q, q^5) / f(q, q^3) in powers of q where f(,) is the two-variable Ramanujan theta function.
%F A143067 Expansion of psi(-q^3) / f(-q^4) in powers of q where psi(), f() are Ramanujan theta functions.
%F A143067 Euler transform of period 12 sequence [ 0, 0, -1, 1, 0, 0, 0, 1, -1, 0, 0, 0, ...].
%F A143067 G.f.: (1 + x + x^5 + x^8 + x^16 + x^21 + ...) / (1 + x + x^3 + x^6 + x^10 + ...). [Ramanujan]
%F A143067 G.f.: 1 - x^3 * (1 - x) / (1 - x^4) + x^8 * (1 - x) * (1 - x^3) / ((1 - x^4) * (1 - x^8)) - ... [Ramanujan]
%e A143067 q^5 - q^77 + q^101 - q^173 + 2*q^197 - q^221 - 2*q^269 + 3*q^293 + ...
%o A143067 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^12 + A) / (eta(x^4 + A) * eta(x^6 + A)), n))}
%Y A143067 Adjacent sequences: A143064 A143065 A143066 this_sequence A143068 A143069 A143070
%Y A143067 Sequence in context: A104041 A104402 A131084 this_sequence A123949 A004718 A055347
%K A143067 sign
%O A143067 0,9
%A A143067 Michael Somos, Jul 21 2008

%I A123949
%S A123949 1,1,1,1,2,1,0,2,3,1,1,0,2,0,1,0,2,1,3,1,1,0,0,3,6,2,2,1,0,2,9,15,11,3,1,
%T A123949 1,1,4,2,6,1,6,1,2,1,0,2,7,1,11,3,8,4,1,1,0,0,3,6,4,18,9,2,3,0,1,0,0,0,
%U A123949 4,3,19,29,11,2,2,1,1,0,0,0,0,4,0,25,16,26,20,4,2,1,0,0,0,4,11,7,63,63,8,34
%V A123949 1,1,-1,1,-2,1,0,-2,3,-1,1,0,-2,0,1,0,-2,-1,3,1,-1,0,0,-3,6,-2,-2,1,0,2,-9,15,-11,3,1,
%W A123949 -1,1,-4,2,6,-1,-6,-1,2,1,0,-2,7,-1,-11,-3,8,4,-1,-1,0,0,-3,-6,4,18,-9,-2,-3,0,1,0,0,0,
%X A123949 -4,3,19,-29,11,-2,2,1,-1,0,0,0,0,4,0,-25,16,26,-20,-4,2,1,0,0,0,-4,11,7,-63,63,8,-34
%N A123949 n-th level Hadamard matrices for Pascal type binary are inverted and multiplied by the corresponding n th level Gray code Hadamard ( some of which matrices have determinant zero): the resulting matrix is processed to give a triangular sequence.
%C A123949 Matrices: 1 X 1 {{1}} 2 X 2 {{1, 0}, {0, 1}} 3 X 3 {{1, -1, -1}, {0, 1, 1}, {0, 1, 1}} 4 X 4 {{1, 1, 0, 0}, {0, -1, 0, 0}, {0, -1, 0, 1}, {0, 2, 1, 0}} 5 X 5 {{1, 1, -1, -1, -1}, {0, -1, 0, 0, 0}, {0, -1, 0, 1, 1}, {0, 2, 1, 0, 0}, {0, 0, 1, 1, 1}} 6 X 6 {{1, 1, 0, -1, -1, 0}, {0, -1, -1, 0, 0, -1}, {0, -1, 0, 1, 1, 0}, {0, 2, 1, 0, 0, 1}, {0, 0, 0, 1, 1, 0}. {0, 0, 1, 0, 0, 1}} They don't get interesting until 4 X 4!
%F A123949 x(i,j)=a(i,j)^(-1).b(i,j) p(n,x)=CharacteristicPolyynomial(x(i,j)) p(n,x)->t(n,m)
%e A123949 Triangular sequence:
%e A123949 {1},
%e A123949 {1, -1},
%e A123949 {1, -2, 1},
%e A123949 {0, -2, 3, -1},
%e A123949 {1, 0, -2, 0, 1},
%e A123949 {0, -2, -1, 3,1, -1},
%e A123949 {0, 0, -3, 6, -2, -2, 1},
%e A123949 {0, 2, -9, 15, -11,3, 1, -1},
%e A123949 {1, -4, 2, 6, -1, -6, -1, 2, 1},
%e A123949 {0, -2,7, -1, -11, -3, 8, 4, -1, -1},
%e A123949 {0, 0, -3, -6, 4, 18, -9, -2, -3, 0, 1}
%e A123949 Polynomials:
%e A123949 1,
%e A123949 1 - x,
%e A123949 1 - 2 x + x^2,
%e A123949 0 -2x + 3x^2 - x^3,
%e A123949 1 +0x - 2x^2 + x^4,
%e A123949 0-2x - x^2 + 3 x^3 + x^4 - x^5,
%e A123949 0+0x +3x^2 + 6 x^3 - 2 x^4 - 2 x^5 + x^6,
%e A123949 0+ 2x - 9 x^2 + 15x^3 - 11 x^4 + 3 x^5 + x^6 -x^7,
%e A123949 1 - 4 x + 2x^2 + 6x^3 - x^4 - 6 x^5 - x^6 + 2 x^7 + x^8
%t A123949 c[i_, k_] := Floor[Mod[i/2^k, 2]]; b[i_, k_] := If[c[i, k] == 0 && c[ i, k + 1] == 0, 0, If[c[i, k] == 1 && c[i, k + 1] == 1, 0, 1]]; An[d_] := Table[If[Sum[b[n, k]*b[m, k], {k, 0, d - 1}] == 0, 1, 0], {n, 0, d - 1}, {m, 0, d - 1}]; Bn[d_] := Table[If[Sum[c[n, k]*c[ m, k], {k, 0, d - 1}] == 0, 1, 0], {n, 0, d - 1}, {m, 0, d - 1}]; Xn[d_] := MatrixPower[Bn[d], -1].An[d]; a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[Xn[d], x], x], {d, 1, 20}]]; Flatten[%]
%Y A123949 Cf. A122944, A121801, A122947.
%Y A123949 Adjacent sequences: A123946 A123947 A123948 this_sequence A123950 A123951 A123952
%Y A123949 Sequence in context: A104402 A131084 A143067 this_sequence A004718 A055347 A055288
%K A123949 uned,probation,tabl,sign
%O A123949 1,5
%A A123949 Gary Adamson and Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 26 2006

%I A004718
%S A004718 0,1,1,2,1,0,2,3,1,2,0,1,2,1,3,4,1,0,2,3,0,1,1,2,2,3,1,0,3,
%T A004718 2,4,5,1,2,0,1,2,1,3,4,0,1,1,2,1,0,2,3,2,1,3,4,1,2,0,1,3,
%U A004718 4,2,1,4,3,5,6,1,0,2,3,0,1,1,2,2,3,1,0,3,2,4,5,0,1,1,2,1,0
%V A004718 0,1,-1,2,1,0,-2,3,-1,2,0,1,2,-1,-3,4,1,0,-2,3,0,1,-1,2,-2,3,1,0,3,
%W A004718 -2,-4,5,-1,2,0,1,2,-1,-3,4,0,1,-1,2,1,0,-2,3,2,-1,-3,4,-1,2,0,1,-3,
%X A004718 4,2,-1,4,-3,-5,6,1,0,-2,3,0,1,-1,2,-2,3,1,0,3,-2,-4,5,0,1,-1,2,1,0
%N A004718 The Danish composer Per Norgard [N&oslash; rg&aring; rd]'s "infinity sequence", invented in an attempt to unify in a perfect way repetition and variation: a(2n) = -a(n), a(2n+1) = a(n) + 1, a(0)=0.
%C A004718 Minima are at n=2^i-2, maxima at 2^i-1, zeros at A083866.
%C A004718 a(n) has parity of Thue-Morse sequence on {0,1} (A010060).
%C A004718 a(n) = A000120(n) for all n in A060142.
%C A004718 The composer Per Norgard's name is also written in the OEIS as Per Noergaard.
%D A004718 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
%H A004718 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b004718.txt">First 10000 terms</a>
%H A004718 J.-P. Allouche, J. Shallit, <a href="http://www.lri.fr/~allouche/kreg2.ps">The Ring of k-regular Sequences, II</a>
%H A004718 Per Noergaard [Norgard], <a href="http://www.pernoergaard.dk/indexeng.html">Home Page</a>
%H A004718 Per Noergaard [Norgard], <a href="http://www.pernoergaard.dk/eng/udvalgte/111b.html">"Voyage into the golden screen"</a>, 2nd movement
%H A004718 Per Noergaard [Norgard], <a href="http://www.pernoergaard.dk/ress/musexx/m1110356.mp3">"Voyage into the golden screen" (MP3 Recording)</a>
%H A004718 Per Noergaard [Norgard], <a href="http://www.pernoergaard.dk/ress/musexx/mu01.mp3">First 128 notes of the infinity series (MP3 Recording)</a>
%H A004718 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0307027">Divide-and-conquer generating functions. I. Elementary sequences</a>
%F A004718 Write n in binary and read from left to write, starting with 0 and interpreting 1 as "add 1" and 0 as "change sign". For example 19 = binary 10011, giving 0 -> 1 -> -1 -> 1 -> 2 -> 3, so a(19) = 3.
%F A004718 G.f.: sum{k>=0, x^(2^k)/[1-x^(2*2^k)] * prod{l=0, k-1, x^(2^l)-1}}.
%F A004718 The g.f. satisfies F(x^2)(1-x) = F(x)-x/(1-x^2).
%p A004718 f:=proc(n) option remember; if n=0 then RETURN(0); fi; if n mod 2 = 0 then RETURN(-f(n/2)); else RETURN(f((n-1)/2)+1); fi; end;
%Y A004718 Adjacent sequences: A004715 A004716 A004717 this_sequence A004719 A004720 A004721
%Y A004718 Sequence in context: A131084 A143067 A123949 this_sequence A055347 A055288 A111374
%K A004718 sign,nice
%O A004718 0,4
%A A004718 Jorn B. Olsson (olsson(AT)math.ku.dk)
%E A004718 Edited by Ralf Stephan (ralf(AT)ark.in-berlin.de), Mar 07 2003

%I A055347
%S A055347 1,1,1,1,2,1,0,2,3,1,3,2,4,4,1,4,13,9,4,5,1,25,2,53,30,5,6,1,98,222,85,
%T A055347 104,69,2,7,1,543,192,1462,644,212,136,0,8,1,4310,8809,2469,4541,2573,
%U A055347 164,242,8,9,1,26980,17664,83744,32573
%V A055347 1,-1,1,1,-2,1,0,2,-3,1,-3,2,4,-4,1,4,-13,9,4,-5,1,25,-2,-53,30,5,-6,
%W A055347 1,-98,222,-85,-104,69,2,-7,1,-543,-192,1462,-644,-212,136,0,-8,1,
%X A055347 4310,-8809,2469,4541,-2573,-164,242,-8,-9,1,26980,17664,-83744,32573
%N A055347 Matrix inverse of triangle A055340(n+1,k).
%H A055347 <a href="http://www.research.att.com/~njas/sequences/Sindx_Mo.html#mobiles">Index entries for sequences related to mobiles</a>
%e A055347 1; -1,1; 1,-2,1; 0,2,-3,1; -3,2,4,-4,1; ...
%Y A055347 Adjacent sequences: A055344 A055345 A055346 this_sequence A055348 A055349 A055350
%Y A055347 Sequence in context: A143067 A123949 A004718 this_sequence A055288 A111374 A072739
%K A055347 sign,tabl
%O A055347 1,5
%A A055347 Christian G. Bower (bowerc(AT)usa.net), May 14 2000

%I A055288
%S A055288 1,1,1,1,2,1,0,2,3,1,3,2,4,4,1,5,11,4,6,5,1,9,4,26,9,9,6,1,44,72,3,52,
%T A055288 15,12,7,1,50,82,213,20,94,24,16,8,1,545,857,53,546,82,146,35,20,9,1,
%U A055288 572,1636,3370,256,1288,216,232,50,25,10
%V A055288 1,-1,1,1,-2,1,0,2,-3,1,-3,2,4,-4,1,5,-11,4,6,-5,1,9,4,-26,9,9,-6,1,
%W A055288 -44,72,3,-52,15,12,-7,1,-50,-82,213,-20,-94,24,16,-8,1,545,-857,-53,
%X A055288 546,-82,-146,35,20,-9,1,572,1636,-3370,256,1288,-216,-232,50,25,-10
%N A055288 Matrix inverse of triangle A055277(n+1,k).
%H A055288 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%e A055288 1; -1,1; 1,-2,1; 0,2,-3,1; -3,2,4,-4,1; ...
%Y A055288 Adjacent sequences: A055285 A055286 A055287 this_sequence A055289 A055290 A055291
%Y A055288 Sequence in context: A123949 A004718 A055347 this_sequence A111374 A072739 A030399
%K A055288 sign,tabl
%O A055288 1,5
%A A055288 Christian G. Bower (bowerc(AT)usa.net), May 09 2000

%I A111374
%S A111374 1,1,1,0,0,1,1,0,1,2,1,0,2,3,2,0,3,4,4,0,4,6,5,0,5,9,6,0,8,12,9,0,12,16,
%T A111374 13,0,14,22,17,0,18,29,21,0,26,38,28,0,34,50,39,0,42,64,49,0,53,82,60,0,
%U A111374 70,105,78,0,90,132,101,0,110,166,125,0,137,208,153,0,174,258,192,0,217
%V A111374 1,1,1,0,0,-1,-1,0,1,2,1,0,-2,-3,-2,0,3,4,4,0,-4,-6,-5,0,5,9,6,0,-8,-12,-9,0,12,16,
%W A111374 13,0,-14,-22,-17,0,18,29,21,0,-26,-38,-28,0,34,50,39,0,-42,-64,-49,0,53,82,60,0,
%X A111374 -70,-105,-78,0,90,132,101,0,-110,-166,-125,0,137,208,153,0,-174,-258,-192,0,217
%N A111374 Series expansion of the Goellnitz-Gordon continued fraction 1 + q + q^2/(1 + q^3 + q^4/(1 + q^5 + q^6/(1 + q^7+ ...))).
%D A111374 S.-D. Chen and S.-S. Huang, On the series expansion of the Goellnitz-Gordon continued fraction, Internat. J. Number Theory, 1 (2005), 53-63.
%F A111374 Let qf(a, q) = Product(1-a*q^j, j=0..infinity); g.f. is qf(q^3, q^8)*qf(q^5, q^8)/(qf(q, q^8)*qf(q^7, q^8)).
%F A111374 Expansion of (phi(q)+phi(q^2))/(2*psi(q^4)) = 2*q*psi(q^4)/(phi(q)-phi(q^2)) in powers of q where phi(),psi() are Ramanujan theta functions. - Michael Somos Feb 15 2006
%e A111374 1/q +q +q^3 -q^9 -q^11 +q^15 +2*q^17 +q^19 -2*q^23 +...
%p A111374 M:=100; qf:=(a,q)->mul(1-a*q^j,j=0..M); t2:=qf(q^3,q^8)*qf(q^5,q^8)/(qf(q,q^8)*qf(q^7,q^8)); series(%,q,M); seriestolist(%);
%Y A111374 Cf. A003823. G.f. is reciprocal of that of A092869.
%Y A111374 Adjacent sequences: A111371 A111372 A111373 this_sequence A111375 A111376 A111377
%Y A111374 Sequence in context: A004718 A055347 A055288 this_sequence A072739 A030399 A128763
%K A111374 sign
%O A111374 0,10
%A A111374 njas, Nov 09 2005

%I A072739
%S A072739 0,0,1,1,2,1,0,2,3,2,1,3,4,3,2,0,2,4,5,4,3,1,3,5,6,5,4,3,0,2,4,6,7,6,5,
%T A072739 4,1,3,5,7,8,7,6,5,4,0,2,4,6,8,9,8,7,6,5,1,3,5,7,9,10,9,8,7,6,5,0,2,4,
%U A072739 6,8,10,11,10,9,8,7,6,1,3,5,7,9,11,12,11,10,9,8,7,6,0,2,4,6,8,10,12,13
%N A072739 Y-projection of the tabular N X N -> N bijection A072733.
%o A072739 (Scheme) (define (A072739 n) (A002262 (A072732 n)))
%Y A072739 The X-projection is A072738. Composition of A002262 and A072732. A072786(n) = A072782(n)-A072739(n).
%Y A072739 Adjacent sequences: A072736 A072737 A072738 this_sequence A072740 A072741 A072742
%Y A072739 Sequence in context: A055347 A055288 A111374 this_sequence A030399 A128763 A127597
%K A072739 nonn,tabl
%O A072739 0,5
%A A072739 Antti Karttunen Jun 12 2002

%I A030399
%S A030399 2,1,0,2,3,2,2,2,1,2,0,1,3,1,2,1,1,1,0,0,3,0,2,0,1,0,0,2,3,3,
%T A030399 2,3,2,2,3,1,2,3,0,2,2,3,2,2,2,2,2,1,2,2,0,2,1,3,2,1,2,2,1,1,
%U A030399 2,1,0,2,0,3,2,0,2,2,0,1,2,0,0,1,3,3,1,3,2,1,3,1,1,3,0,1,2,3
%N A030399 Write n in base 4, then complement each digit (d -> 3-d) and juxtapose.
%Y A030399 Adjacent sequences: A030396 A030397 A030398 this_sequence A030400 A030401 A030402
%Y A030399 Sequence in context: A055288 A111374 A072739 this_sequence A128763 A127597 A104770
%K A030399 nonn
%O A030399 1,1
%A A030399 Clark Kimberling (ck6(AT)evansville.edu)

%I A128763
%S A128763 1,1,0,1,2,1,0,2,3,2,2,4,6,5,4,6,9,8,6,10,15,14,12,17,24,21,18,26,35,32,
%T A128763 30,42,52,50,48,60,75,74,70,88,111,109,104,130,158,154,150,184,220,218,
%U A128763 218,262,308,308,308,362,421,426,428,498,580,589,592,685,788,796
%V A128763 1,-1,0,-1,2,-1,0,-2,3,-2,2,-4,6,-5,4,-6,9,-8,6,-10,15,-14,12,-17,24,-21,18,-26,35,-32,
%W A128763 30,-42,52,-50,48,-60,75,-74,70,-88,111,-109,104,-130,158,-154,150,-184,220,-218,218,
%X A128763 -262,308,-308,308,-362,421,-426,428,-498,580,-589,592,-685,788,-796
%N A128763 Expansion of chi(q^5)* chi(q^10)/( chi(q)* chi(q^2)) in powers of q where chi() is a Ramanujan theta function.
%F A128763 Euler transform of period 40 sequence [ -1, 0, -1, 1, 0, 0, -1, 0, -1, 0, -1, 1, -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, 0, 0, -1, 1, -1, 0, -1, 0, -1, 0, 0, 1, -1, 0, -1, 0, ...].
%F A128763 Given g.f. A(x), then B(x)= 1/x*A(x^2) satisfies 0= f(B(x), B(x^3)) where f(u, v)= (u-v^3)* (u^3-v) -3*u*v* (u^2+v^2).
%F A128763 G.f.: Product_{k>0} (1+x^(4k))* (1+x^(5k))/( (1+x^k)* (1+x^(20k)) ).
%e A128763 1/q - q - q^5 + 2*q^7 - q^9 - 2*q^13 + 3*q^15 - 2*q^17 + 2*q^19 - ...
%o A128763 (PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( eta(x+A)* eta(x^8+A)* eta(x^10+A)* eta(x^20+A)/ (eta(x^2+A)* eta(x^4+A)* eta(x^5+A)* eta(x^40+A)), n))}
%Y A128763 Convolution inverse of A128762.
%Y A128763 Adjacent sequences: A128760 A128761 A128762 this_sequence A128764 A128765 A128766
%Y A128763 Sequence in context: A111374 A072739 A030399 this_sequence A127597 A104770 A110280
%K A128763 sign
%O A128763 0,5
%A A128763 Michael Somos, Mar 25 2007

%I A127597
%S A127597 2,1,0,2,3,2,4,4,3,10,3,3,2,7,2,25,6,17,4,13,3,20,36,20,11,27,66,23,39,
%T A127597 24,19,13,3,10,6,122,71,58,24,13,3,2,41,10,6,32,58,17,4,79,26
%N A127597 Least number k such that k 4^n + (4^n-1)/3 is prime.
%t A127597 a = {}; Do[k = 0; While[ !PrimeQ[k 4^n + (4^n - 1)/3], k++ ]; AppendTo[a, k], {n, 0, 50}]; a (*Artur Jasinski*)
%Y A127597 Cf. A035050, A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587, A127589, A127590, A127591, A127592, A127593, A127594, A127598.
%Y A127597 Adjacent sequences: A127594 A127595 A127596 this_sequence A127598 A127599 A127600
%Y A127597 Sequence in context: A072739 A030399 A128763 this_sequence A104770 A110280 A061009
%K A127597 nonn
%O A127597 1,1
%A A127597 Artur Jasinski (grafix(AT)csl.pl), Jan 19 2007

%I A104770
%S A104770 1,1,2,1,0,2,3,3,1,2,5,6,4,1,7,11,10,3,8,18,21,13,5,26,39,34,8,31,65,73,
%T A104770 42,23,96,138,115,19,119,234,253,134,100,353,487,387,34,453,840,874,421,
%U A104770 419,1293,1714,1295,2,1712,3007,3009,1297,1710,4719,6016
%V A104770 1,-1,2,-1,0,2,-3,3,-1,-2,5,-6,4,1,-7,11,-10,3,8,-18,21,-13,-5,26,-39,34,-8,-31,65,-73,
%W A104770 42,23,-96,138,-115,19,119,-234,253,-134,-100,353,-487,387,-34,-453,840,-874,421,419,
%X A104770 -1293,1714,-1295,2,1712,-3007,3009,-1297,-1710,4719,-6016
%N A104770 G.f. -(x^2+1)/(x^3-x-1).
%C A104770 A floretion-generated sequence which may be seen as the result of a certain sequence transform applied infinitely often. For a related case, see "Generalized Sequence Convergence?" link. This sequence is one of three related sequences, the others being A104771 and A104769.
%F A104770 a(n+3) = a(n) - a(n+2); a(0) = 1, a(1) = -1, a(2) = 2; a(n+1) - a(n) = ((-1)^(n+1))*a(n+5); a(n) = A104771(n) - A104769(n)
%o A104770 Floretion Algebra Multiplication Program, FAMP Code: Define A = + .5'i + .5'j + .5'k + .5e and B = + .5'i + .5i' + .5'ii' + .5e. Then (a(n)) = jesloop(infty)-jesleftfor[A*B], ForType: 1A.
%Y A104770 Cf. A104769, A104771.
%Y A104770 Adjacent sequences: A104767 A104768 A104769 this_sequence A104771 A104772 A104773
%Y A104770 Sequence in context: A030399 A128763 A127597 this_sequence A110280 A061009 A104558
%K A104770 sign,uned
%O A104770 0,3
%A A104770 Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Mar 24 2005

%I A110280
%S A110280 1,1,2,1,0,2,3,3,2,4,5,11,10,7,4,6,7,13,10,3,1,12,12,17,15,8,3,7,10,14,
%T A110280 11,4,1,13,14,20,11,11,4,11,13,24,7,11,8,16,19,29,13,14,7,13,18,26,8,9,
%U A110280 11,22,22,35,15,15,7,20,24,35,17,13,11,26,34,43,27,19,7,24,36,43,29,17,11
%V A110280 1,1,2,1,0,-2,-3,-3,-2,4,5,11,10,7,4,-6,-7,-13,-10,-3,-1,12,12,17,15,8,3,-7,-10,-14,
%W A110280 -11,-4,1,13,14,20,11,11,-4,-11,-13,-24,-7,-11,8,16,19,29,13,14,-7,-13,-18,-26,-8,-9,
%X A110280 11,22,22,35,15,15,-7,-20,-24,-35,-17,-13,11,26,34,43,27,19,-7,-24,-36,-43,-29,-17,11
%N A110280 A floretion-generated sequence calculated using the same rules given for A108618 with initial seed x = + .5'i + .5'ii' + .5'ij' + .5'ik'; version: "ibase".
%C A110280 The initial seed + .5'i + .5'ii' + .5'ij' + .5'ik' can be seen as an element of the space Q X C_3 where Q are the quaternions.
%o A110280 Floretion Algebra Multiplication Program, FAMP Code: 2ibasesumseq[ + .5'i + .5'ii' + .5'ij' + .5'ik']. SumType is set to: sum[Y[15]]
%Y A110280 Cf. A108618, A110279, A110281, A110282, A110283.
%Y A110280 Adjacent sequences: A110277 A110278 A110279 this_sequence A110281 A110282 A110283
%Y A110280 Sequence in context: A128763 A127597 A104770 this_sequence A061009 A104558 A115247
%K A110280 easy,sign
%O A110280 0,3
%A A110280 Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Jul 18 2005

%I A061009
%S A061009 2,1,0,2,3,3,4,4,4,4,5,5,6,6,6,6,7,7,8,8,8,8,9,9,9,9,9,9,10,10,11,11,11,11,11,11,12,
%T A061009 12,12,12,13,13,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,18,18,18,18,18,
%U A061009 18,19,19,19,19,20,20,21,21,21,21,21,21,22,22,22,22,23,23,23,23,23,23,24,24,24,24,24
%V A061009 -2,-1,0,2,3,3,4,4,4,4,5,5,6,6,6,6,7,7,8,8,8,8,9,9,9,9,9,9,10,10,11,11,11,11,11,11,12,
%W A061009 12,12,12,13,13,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,18,18,18,18,18,
%X A061009 18,19,19,19,19,20,20,21,21,21,21,21,21,22,22,22,22,23,23,23,23,23,23,24,24,24,24,24
%N A061009 -2+sum_j (-(n-1)! mod n) over 0<j<=n.
%F A061009 a(n) =a(n-1)+A061007(n) =A061008(n)-2. For n>3, a(n)=pi(n)=A000720(n) where pi(n) is the number of primes less than or equal to n.
%e A061009 a(6)=3 since -2 + (-1 mod 1) + (-1 mod 2) + (-2 mod 3) + (-6 mod 4) + (-24 mod 5) + (-120 mod 6) = -2+0+1+1+2+1+0 = 3.
%p A061009 P:=proc(n) local a,i,w; a:=-2; print(a); for i from 1 by 1 to n do w:=((i-1)! mod (i+1)); a:=a+w; print(a); od; end: P(1000); - Paolo P. Lava (ppl(AT)spl.at), Apr 23 2007
%Y A061009 Cf. A000040, A000142, A061006, A061007, A061008.
%Y A061009 Adjacent sequences: A061006 A061007 A061008 this_sequence A061010 A061011 A061012
%Y A061009 Sequence in context: A127597 A104770 A110280 this_sequence A104558 A115247 A122542
%K A061009 sign
%O A061009 1,1
%A A061009 Henry Bottomley (se16(AT)btinternet.com), Apr 12 2001

%I A104558
%S A104558 1,1,1,0,2,1,0,2,4,1,0,0,6,6,1,0,0,6,18,9,1,0,0,0,24,36,12,1,0,0,0,24,96,72,
%T A104558 16,1,0,0,0,0,120,240,120,20,1,0,0,0,0,120,600,600,200,25,1,0,0,0,0,0,720,1800,
%U A104558 1200,300,30,1,0,0,0,0,0,720,4320,5400,2400,450,36,1,0,0,0,0,0,0,5040,15120
%V A104558 1,-1,1,0,-2,1,0,2,-4,1,0,0,6,-6,1,0,0,-6,18,-9,1,0,0,0,-24,36,-12,1,0,0,0,24,-96,72,
%W A104558 -16,1,0,0,0,0,120,-240,120,-20,1,0,0,0,0,-120,600,-600,200,-25,1,0,0,0,0,0,-720,1800,
%X A104558 -1200,300,-30,1,0,0,0,0,0,720,-4320,5400,-2400,450,-36,1,0,0,0,0,0,0,5040,-15120
%N A104558 Triangle, read by rows, equal to the matrix inverse of A104557, and related to Laguerre polynomials.
%C A104558 Even-indexed rows are found in A066667 (generalized Laguerre polynomials). Odd-indexed rows are found in A021009 (Laguerre polynomials L_n(x)). Row sums equal A056920 (offset 1). Absolute row sums equal A056953 (offset 1).
%F A104558 T(n, k) = (-1)^(n-k)*(n-k)!*C(1+[n/2], k+1-[(n+1)/2])*C([(n+1)/2], k-[n/2]).
%e A104558 Rows begin:
%e A104558 1;
%e A104558 -1,1;
%e A104558 0,-2,1;
%e A104558 0,2,-4,1;
%e A104558 0,0,6,-6,1;
%e A104558 0,0,-6,18,-9,1;
%e A104558 0,0,0,-24,36,-12,1;
%e A104558 0,0,0,24,-96,72,-16,1;
%e A104558 0,0,0,0,120,-240,120,-20,1;
%e A104558 0,0,0,0,-120,600,-600,200,-25,1; ...
%e A104558 Unsigned columns read downwards equals rows of
%e A104558 matrix inverse A104557 read backwards:
%e A104558 1;
%e A104558 1,1;
%e A104558 2,2,1;
%e A104558 6,6,4,1;
%e A104558 24,24,18,6,1;
%e A104558 120,120,96,36,9,1; ...
%o A104558 (PARI) {T(n,k)=(-1)^(n-k)*(n-k)!*binomial(1+n\2,k+1-(n+1)\2)*binomial((n+1)\2,k-n\2)}
%Y A104558 Cf. A104557, A066667, A021009, A056920, A056953.
%Y A104558 Adjacent sequences: A104555 A104556 A104557 this_sequence A104559 A104560 A104561
%Y A104558 Sequence in context: A104770 A110280 A061009 this_sequence A115247 A122542 A098542
%K A104558 sign,tabl
%O A104558 0,5
%A A104558 Paul D. Hanna (pauldhanna(AT)juno.com), Mar 16 2005

%I A115247
%S A115247 0,0,1,0,2,1,0,2,4,1,0,1,1,6,1,0,2,1,4,1,6,0,2,4,1,2,2,8,0,1,1,2,1,4,6,
%T A115247 1,0,3,4,1,2,7,1,6,1,0,1,4,6,2,1,1,2,1,1,0,1,1,8,2,4,2,6,4,3,1,0,2,4,7,
%U A115247 3,1,1,4,1,1,6,1,0,2,1,2,1,6,1,2,4,2,7,8,6,0,2,4,2,1,1,6,4,4
%N A115247 2^a(n) divides A001935(n) but 2^(a(n)+1) does not.
%C A115247 Almost all members of A001935 are divisible by 2^k for any k, therefore almost all a(n)>k for any k.
%H A115247 Basil Gordon and Ken Ono, <a href="http://www.math.wisc.edu/~ono/reprints/018.pdf">Divisibility of Certain Partition Functions By Powers of Primes</a>.
%H A115247 K. Alladi, <a href="http://www.ams.org/tran/1997-349-12/S0002-9947-97-01831-X/S0002-9947-97-01831-X.pdf">Partition Identities Involving Gaps and Weights</a>, Transactions of the American Mathematical Society, Vol. 349, No. 12, Dec 1997, pp. 5001-5019.
%Y A115247 The 0's are in A000217. The 1's are in A115248. Least inverse A115250.
%Y A115247 Adjacent sequences: A115244 A115245 A115246 this_sequence A115248 A115249 A115250
%Y A115247 Sequence in context: A110280 A061009 A104558 this_sequence A122542 A098542 A141343
%K A115247 nonn
%O A115247 0,5
%A A115247 Christian G. Bower (bowerc(AT)usa.net), Jan 17 2006

%I A122542
%S A122542 1,0,1,0,2,1,0,2,4,1,0,2,8,6,1,0,2,12,18,8,1,0,2,16,38,32,10,1,0,2,20,
%T A122542 66,88,50,12,1,0,2,24,102,192,170,72,14,1,0,2,28,146,360,450,292,98,16,
%U A122542 1,0,2,32,198,608,1002,912,462,128,18,1
%N A122542 Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the oprator defined in A084938.
%C A122542 Riordan array (1, x*(1+x)/(1-x)) . Rising and falling diagonals are the tribonacci numbers A000213, A001590.
%F A122542 Sum_{k, 0<=k<=n}x^k*T(n,k) = A001333(n), A104934(n) for x=1, 2 . Sum_{k, 0<=k<=n}3^(n-k)*T(n,k) = A086901(n).
%F A122542 Sum_{k, 0<=k<=n}2^(n-k)*T(n,k)=A007483(n-1), n>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 08 2006
%F A122542 T(2*n,n)=A123164(n+1).
%e A122542 Triangle begins:
%e A122542 1;
%e A122542 0, 1;
%e A122542 0, 2, 1;
%e A122542 0, 2, 4, 1;
%e A122542 0, 2, 8, 6, 1;
%e A122542 0, 2, 12, 18, 8, 1;
%e A122542 0, 2, 16, 38, 32, 10, 1;
%e A122542 0, 2, 20, 66, 88, 50, 12, 1;
%e A122542 0, 2, 24, 102, 192, 170, 72, 14, 1;
%e A122542 0, 2, 28, 146, 360, 450, 292, 98, 16, 1;
%e A122542 0, 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1;
%Y A122542 Cf. A113413, A035607. Diagonals : A000012, A005843, A001105, A035597-A035606. Columns : A000007, A040000, A008575, A005899, A008412-A008416, A008418, A008420, A035706-A035745.
%Y A122542 Adjacent sequences: A122539 A122540 A122541 this_sequence A122543 A122544 A122545
%Y A122542 Sequence in context: A061009 A104558 A115247 this_sequence A098542 A141343 A066709
%K A122542 nonn,tabl
%O A122542 0,5
%A A122542 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 19 2006, May 28 2007

%I A098542
%S A098542 1,1,1,0,2,1,0,2,4,1,0,2,12,8,1,0,2,44,56,16,1,0,2,236,504,240,32,1,0,2,
%T A098542 2028,6776,4720,992,64,1,0,2,29164,146552,139120,40672,4032,128,1,0,2,
%U A098542 719340,5314680,6583152,2500832,337344,16256,256,1,0,2,30943724
%N A098542 Triangle T, read by rows, such that the matrix square shifts T left one column and up one row, with T(0,0)=T(1,0)=1 and T(n,0)=0 for n>1, and T(n,n)=1 for n>=0.
%C A098542 Column 2 forms A098543. Row sums form A098544. The absolute value of the matrix inverse equals A098539.
%F A098542 T(n+1, 1) = 2 for n>0; T(n+1, n) = 2^n, T(n+2, n) = 4^n - 2^n for n>=0. Matrix square: [T^2](n, k) = T(n+1, k+1). Matrix inverse: [T^-1](n, k) = (-1)^(n-k)*A098539(n, k). Matrix square inverse: [T^-2](n, k) = (-1)^(n-k)*A098539(n+1, k+1).
%e A098542 Rows of T begin:
%e A098542 [1],
%e A098542 [1,1],
%e A098542 [0,2,1],
%e A098542 [0,2,4,1],
%e A098542 [0,2,12,8,1],
%e A098542 [0,2,44,56,16,1],
%e A098542 [0,2,236,504,240,32,1],
%e A098542 [0,2,2028,6776,4720,992,64,1],
%e A098542 [0,2,29164,146552,139120,40672,4032,128,1],
%e A098542 [0,2,719340,5314680,6583152,2500832,337344,16256,256,1],...
%e A098542 Rows of T^2 begin:
%e A098542 [1],
%e A098542 [2,1],
%e A098542 [2,4,1],
%e A098542 [2,12,8,1],
%e A098542 [2,44,56,16,1],
%e A098542 [2,236,504,240,32,1],...
%e A098542 showing that T shifts left and up under matrix square.
%e A098542 The matrix inverse of T begins:
%e A098542 [1],
%e A098542 [ -1,1],
%e A098542 [2,-2,1],
%e A098542 [ -6,6,-4,1],
%e A098542 [26,-26,20,-8,1],
%e A098542 [ -166,166,-140,72,-16,1],...
%e A098542 the absolute value of which equals triangle A098539.
%o A098542 (PARI) {T(n,k)=local(A,B);A=matrix(1,1);A[1,1]=1;for(m=2,n+1,B=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3|j==i|j>m-1,B[i,j]=1,if(j==1, B[i,j]=(A^0)[i-1,1],B[i,j]=(A^2)[i-1,j-1]));));A=B);A[n+1,k+1]}
%Y A098542 Cf. A098543, A098544, A098539 (absolute inverse).
%Y A098542 Adjacent sequences: A098539 A098540 A098541 this_sequence A098543 A098544 A098545
%Y A098542 Sequence in context: A104558 A115247 A122542 this_sequence A141343 A066709 A108354
%K A098542 nonn,tabl
%O A098542 0,5
%A A098542 Paul D. Hanna (pauldhanna(AT)juno.com), Sep 16 2004

%I A141343
%S A141343 1,0,1,0,2,1,0,2,4,1,0,4,8,6,1,0,2,16,18,8,1,0,12,24,44,32,10,1,0,12,48,
%T A141343 90,96,50,12,1,0,72,48,180,240,180,72,14,1,0,190,160,308,544,530,304,98,
%U A141343 16,1,0,700,40,600,1120,1372,1032,476,128
%V A141343 1,0,1,0,2,1,0,2,4,1,0,4,8,6,1,0,2,16,18,8,1,0,12,24,44,32,10,1,0,-12,48,90,96,50,12,1,
%W A141343 0,72,48,180,240,180,72,14,1,0,-190,160,308,544,530,304,98,16,1,0,700,-40,600,1120,
%X A141343 1372,1032,476,128
%N A141343 Riordan array (1,x/sqrt(2-sqrt(1+4x)).
%C A141343 Factorises as (1,xc(-x))*(1,x(1+x)/(1-2x)) where c(x) is the g.f. of A000108.
%C A141343 Row sums are A141344.
%e A141343 Triangle begins
%e A141343 1,
%e A141343 0, 1,
%e A141343 0, 2, 1,
%e A141343 0, 2, 4, 1,
%e A141343 0, 4, 8, 6, 1,
%e A141343 0, 2, 16, 18, 8, 1,
%e A141343 0, 12, 24, 44, 32, 10, 1,
%e A141343 0, -12, 48, 90, 96, 50, 12, 1
%Y A141343 Adjacent sequences: A141340 A141341 A141342 this_sequence A141344 A141345 A141346
%Y A141343 Sequence in context: A115247 A122542 A098542 this_sequence A066709 A108354 A118208
%K A141343 easy,sign,tabl
%O A141343 0,5
%A A141343 Paul Barry (pbarry(AT)wit.ie), Jun 26 2008

%I A066709
%S A066709 1,0,1,1,2,1,0,2,4,1,1,5,8,5,1,0,3,14,15,6,1,1,9,25,32,21,7,1,0,4,32,
%T A066709 62,56,28,8,1,1,14,56,109,122,84
%N A066709 Triangle T(r,c) of winning binary "same game" templates.
%C A066709 T(r,c) is the number of winning templates with length r and total r+c of ternary digits. For a definition and row sums 1,1,4,7,20, etc. see A066345. For antidiagonal sums 1,0,2,2,4,9, etc. see A066346. A035615(n)= 2 *sum( r=1 to n-1, c=1 to min(r,n-r): T(r,c) *P(n-r,c)), see A007318 for P(n-r,c)= C(n-r-1,c-1)= (n-r-1)!/((n-r-c-2)!*(c-1)!).
%e A066709 Rows: 1; 0,1; 1,2,2; 0,2,4,1; 1,5,8,5,1; 0,3,14,15,6,1; ...
%e A066709 a(17)= T(6,2)= 3 winning templates with length 6 and total 8= 6+2: 211211, 121121, 112112.
%e A066709 A035615(6)= 2*( 1*1+0*1+1*3+1*1+2*2+1*1+1*1+0*1+2*1+1*1 )= 2*13= 26.
%Y A066709 Cf. A035615, A066345, A066346, A007318.
%Y A066709 Adjacent sequences: A066706 A066707 A066708 this_sequence A066710 A066711 A066712
%Y A066709 Sequence in context: A122542 A098542 A141343 this_sequence A108354 A118208 A074142
%K A066709 nonn,more,tabl
%O A066709 1,5
%A A066709 frank.ellermann(AT)t-online.de, Dec 31 2001

%I A108354
%S A108354 1,2,1,0,2,4,2,0,3,6,3,0,4,8,4,0,5,10,5,0,6,12,6,0,7,14,7,0,8,16,8,0,9,
%T A108354 18,9,0,10,20,10,0,11,22,11,0,12,24,12,0,13,26,13,0,14,28,14,0,15,30,15,
%U A108354 0,16,32,16,0,17,34,17,0,18,36,18,0,19,38,19,0,20,40,20,0,21,42,21,0,22
%N A108354 Expansion of 1/((1-x)^2(1+x^2)^2).
%C A108354 Self-convolution transform of A133872. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 06 2008
%F A108354 a(n)=2a(n-1)-3a(n-2)+4a(n-3)-3a(n-4)+2a(n-5)-a(n-6); a(n)=cos(pi*n/2)/4+(n+3)*sin(pi*n/2)/4+(n+3)/4.
%Y A108354 Adjacent sequences: A108351 A108352 A108353 this_sequence A108355 A108356 A108357
%Y A108354 Sequence in context: A098542 A141343 A066709 this_sequence A118208 A074142 A059084
%K A108354 easy,nonn
%O A108354 0,2
%A A108354 Paul Barry (pbarry(AT)wit.ie), May 31 2005

%I A118208
%S A118208 1,1,2,1,0,2,4,5,3,0,4,6,6,2,3,8,10,6,0,6,14,13,9,0,12,17,18,11,3,18,28,
%T A118208 22,14,7,25,30,31,11,12,23,34,28,9,12,30,35,31,10,11,30,56,35,26,4,41,51,
%U A118208 65,48,8,28,65,74,70,9,49,71,112,69,4,48,135,129,82,21,83,155,176,99,0
%V A118208 1,-1,2,-1,0,2,-4,5,-3,0,4,-6,6,-2,-3,8,-10,6,0,-6,14,-13,9,0,-12,17,-18,11,3,-18,28,
%W A118208 -22,14,7,-25,30,-31,11,12,-23,34,-28,9,12,-30,35,-31,10,11,-30,56,-35,26,-4,-41,51,
%X A118208 -65,48,-8,-28,65,-74,70,-9,-49,71,-112,69,-4,-48,135,-129,82,-21,-83,155,-176,99,0
%N A118208 G.f.: A(x) = product_{k>=1}(1 + x^k)^(-lambda(k)) where lambda(k) is the Liouville function, A008836.
%t A118208 nmax = 80; lambda[k_Integer?Positive] := If[ k > 1, (-1)^Total[ Part[Transpose[FactorInteger[k]], 2] ], 1 ]; CoefficientList[ Series[ Product[ (1 + x^k)^(-lambda[k]), {k, 1, nmax} ], {x, 0, nmax} ], x ]
%Y A118208 Cf. A118205, A118206, A118207, A118209, A117211.
%Y A118208 Adjacent sequences: A118205 A118206 A118207 this_sequence A118209 A118210 A118211
%Y A118208 Sequence in context: A141343 A066709 A108354 this_sequence A074142 A059084 A070677
%K A118208 sign,easy
%O A118208 0,3
%A A118208 Stuart Clary (clary(AT)uakron.edu), Apr 15, 2006

%I A074142
%S A074142 1,1,1,0,0,2,1,0,2,5,2,3,5,6,10,12,9,11,32,11,5,55,61,29,84,129,9,188,
%T A074142 232,136,322,567,255,354,1185,840,585,2038,2318,594,3909,4761,929,7387,
%U A074142 10441,3930,11137,23097,12215,16547,44716,36786,23108
%V A074142 1,-1,-1,0,0,2,1,0,-2,-5,2,3,5,6,-10,-12,-9,11,32,11,-5,-55,-61,29,84,129,9,-188,-232,
%W A074142 -136,322,567,255,-354,-1185,-840,585,2038,2318,-594,-3909,-4761,-929,7387,10441,3930,
%X A074142 -11137,-23097,-12215,16547,44716,36786,-23108
%N A074142 Coefficients a(n) of a series connected with the odd primes.
%C A074142 The series reciprocal to the series with coefficients in A005097 has (integer) coefficients with irregular signs and values. In contrast the series reciprocal to the series with coefficients = primes themselves has coefficients (A030018) with alternating signs and regular growth. The radius of convergence (defined from consecutive coefficients ratio) of that series is 0.686777834460.
%F A074142 Sum(a(i)*x^(i-1), (i=1, inf))=1/(1+sum(1/2(p(i)-1)*x^(i-1), i=2, inf)) =1/(1+sum( A005097(i)*x^(i), i=1, inf))
%Y A074142 Cf. A005097, A030018.
%Y A074142 Adjacent sequences: A074139 A074140 A074141 this_sequence A074143 A074144 A074145
%Y A074142 Sequence in context: A066709 A108354 A118208 this_sequence A059084 A070677 A029584
%K A074142 easy,sign
%O A074142 1,6
%A A074142 Zak Seidov (zakseidov(AT)yahoo.com), Sep 16 2002

%I A059084
%S A059084 1,1,1,2,1,0,2,5,4,1,0,0,12,44,67,56,28,8,1,0,0,12,268,1411,4032,7840,
%T A059084 11392,12864,11440,8008,4368,1820,560,120,16,1,0,0,0,1120,20160,159656,
%U A059084 827092,3251736,10389635,27934400,64432160,128980800,225774640
%N A059084 Triangle T(n,m) of number of labeled n-node T_0-hypergraphs with m distinct hyperedges (empty hyperedge included),m=0,1,...,2^n.
%C A059084 A hypergraph is a T_0 hypergraph if for every two distinct nodes there exists a hyperedge containing one but not the other node.
%H A059084 V. Jovovic, <a href="http://www.research.att.com/~njas/sequences/a59084.pdf">Illustration of initial terms of A059084, A059085</a>
%F A059084 T(n, m)=Sum_{i=0..n} s(n, i)*binomial(2^i, m), where s(n, i) are Stirling numbers of the first kind.
%F A059084 Also T(n, m)=(1/m!)*Sum_{i=0..m} s(m, i)*fallfac(2^i, n). E.g.f: Sum((1+x)^(2^n)*ln(1+y)^n/n!, n=0..infinity). - Vladeta Jovovic (vladeta(AT)Eunet.yu), May 19 2004
%e A059084 [1,1],[1,2,1],[0,2,5,4,1],[0,0,12,44,67,56,28,8,1],...; There are 12 labeled 3-node T_0-hypergraphs with 2 distinct hyperedges: {{3},{2}}, {{3},{2,3}}, {{2},{2,3}}, {{3},{1}}, {{3},{1,3}}, {{2},{1}}, {{2,3},{1,3}}, {{2},{1,2}}, {{2,3},{1,2}}, {{1},{1,3}}, {{1},{1,2}}, {{1,3},{1,2}}.
%Y A059084 Cf. A059085, A059086.
%Y A059084 Cf. A088309.
%Y A059084 Adjacent sequences: A059081 A059082 A059083 this_sequence A059085 A059086 A059087
%Y A059084 Sequence in context: A108354 A118208 A074142 this_sequence A070677 A029584 A136255
%K A059084 easy,nonn,tabf
%O A059084 0,4
%A A059084 Goran Kilibarda, Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 27 2000

%I A070677
%S A070677 0,1,2,1,0,2,6,2,6,0,5,2,4,6,0,4,16,6,9,0,6,5,22,2,0,4,18,6,14,
%T A070677 0,3,8,10,16,0,6,36,9,4,0,20,6,42,5,0,22,46,4,42,0,16,4,52,18,0,
%U A070677 6,18,14,29,0,30,3,6,16,0,10,22,16,22,0,5,6,72,36,0,9,30,4,39,0
%N A070677 Smallest m in range 1..phi(n) such that 5^m == 1 mod n, or 0 if no such number exists.
%Y A070677 Cf. A070667-A070675, A002326, A070676, A053447, A070681, A070678, A053451, A070679, A070682, A070680, A070683.
%Y A070677 Adjacent sequences: A070674 A070675 A070676 this_sequence A070678 A070679 A070680
%Y A070677 Sequence in context: A118208 A074142 A059084 this_sequence A029584 A136255 A127373
%K A070677 nonn
%O A070677 1,3
%A A070677 njas and Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 08 2002

%I A029584
%S A029584 2,1,0,2,6,16,60,272,1386,7936,50520,353792,2702766,22368256,
%T A029584 199360980,1903757312,19391512146,209865342976,2404879675440,
%U A029584 29088885112832,370371188237526,4951498053124096,69348874393137900
%N A029584 Expansion of cos x + tan x + sec x.
%C A029584 a(2n) = 2*A012007(n), a(2n+1) = A000182(n+1).
%Y A029584 Adjacent sequences: A029581 A029582 A029583 this_sequence A029585 A029586 A029587
%Y A029584 Sequence in context: A074142 A059084 A070677 this_sequence A136255 A127373 A050464
%K A029584 nonn
%O A029584 0,1
%A A029584 njas

%I A136255
%S A136255 1,0,2,1,0,3,0,0,0,4,3,0,3,0,5,0,6,0,8,0,6,5,0,6,0,15,0,7,0,16,0,0,0,24,
%T A136255 0,8,7,0,30,0,15,0,35,0,9,0,30,0,40,0,42,0,48,0,10,9,0,75,0,35,0,84,0,
%U A136255 63,0,11
%V A136255 1,0,2,1,0,3,0,0,0,4,-3,0,-3,0,5,0,-6,0,-8,0,6,5,0,-6,0,-15,0,7,0,16,0,0,0,-24,0,8,-7,
%W A136255 0,30,0,15,0,-35,0,9,0,-30,0,40,0,42,0,-48,0,10,9,0,-75,0,35,0,84,0,-63,0,11
%N A136255 Differentiation of:A135929 Triangle read by rows: row n gives coefficients of Differential Boubaker polynomial P(x,n) in order of decreasing exponents.
%C A136255 Row sums are:
%C A136255 Table[Apply[Plus, CoefficientList[P[x, n], x]], {n, 0, 10}]
%C A136255 {1, 2, 4, 4, -1, -8, -9, 0, 12, 14, 1}
%C A136255 Double Integrations are alternating:
%C A136255 Table[Table[Integrate[Sqrt[1/(1 - x^2)]*P[x,n]*P[x, m], {x, -1, 1}], {n, 0, 10}], {m, 0, 10}]
%D A136255 Karem Boubaker, On modified Boubaker polynomials..., Trends in Appl. Sci. Research, 2 (2007), 540-544.
%D A136255 Karem Boubaker et al., Enhancement of pyrolysis spray disposal performance ..., Eur. Phys. J. Appl. Phys., 37 (2007), 105-109. [Link requires a subscription]
%D A136255 Hedi Labiadh and Karem Boubaker, A Sturm-Liouville shaped characteristic differential equation ..., Differential Equations and Control Processes, No. 2 (2007).
%F A136255 Defferentiation of recursive polynomials: B(x, n) = x*B(x, n - 1) - B(x, n - 2); P(x,n)=dB(x,n+1)/dx
%e A136255 {1},
%e A136255 {0, 2},
%e A136255 {1, 0, 3},
%e A136255 {0, 0, 0, 4},
%e A136255 {-3, 0, -3, 0, 5},
%e A136255 {0, -6, 0, -8, 0, 6},
%e A136255 {5, 0, -6, 0, -15, 0, 7},
%e A136255 {0, 16, 0, 0, 0, -24, 0, 8},
%e A136255 {-7, 0, 30, 0, 15, 0, -35, 0, 9},
%e A136255 {0, -30, 0, 40, 0,42, 0, -48, 0, 10},
%e A136255 {9, 0, -75, 0, 35, 0, 84, 0, -63, 0, 11}
%t A136255 Clear[B, x, n] B[x, 0] = 1; B[x, 1] = x; B[x, 2] = 2 + x^2; B[x, 3] = x + x^3; B[x, 4] = -2 + x^4; B[x_, n_] := B[x, n] = x*B[x, n - 1] - B[x, n - 2] P[x_, n_] := D[B[x, n + 1], x] Table[ExpandAll[P[x, n]], {n, 0, 10}] a = Table[CoefficientList[P[x, n], x], {n, 0, 10}] Flatten[a]
%Y A136255 Cf. A138034, A135929, A135936, A137276, A137277, A137289.
%Y A136255 Adjacent sequences: A136252 A136253 A136254 this_sequence A136256 A136257 A136258
%Y A136255 Sequence in context: A059084 A070677 A029584 this_sequence A127373 A050464 A014405
%K A136255 uned,tabl,sign
%O A136255 1,3
%A A136255 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 17 2008

%I A127373
%S A127373 1,1,0,2,1,0,3,0,1,0,6,2,1,1,0,5,0,0,0,1,0,12,4,2,1,1,1,0,13,0,1,0,1,0,
%T A127373 1,0,18,4,0,2,1,0,1,1,0,15,0,3,0,0,0,1,0,1,0
%N A127373 Triangle, row sums = A023896, left column = A053570.
%C A127373 Row sums = A023896: (1, 1, 3, 4, 10, 6,...); left column = A053570: (1, 2, 3, 6, 5, 12,...).
%F A127373 A054521 * A054523 as infinite lower triangular matrices.
%e A127373 First few rows of the triangle are:
%e A127373 1;
%e A127373 1, 0;
%e A127373 2, 1, 0;
%e A127373 3, 0, 1, 0;
%e A127373 6, 2, 1, 1, 0;
%e A127373 5, 0, 0, 0, 1, 0
%e A127373 12, 4, 2, 1, 1, 1, 0;
%e A127373 ...
%Y A127373 Cf. A054521, A054523, A053570, A023896.
%Y A127373 Adjacent sequences: A127370 A127371 A127372 this_sequence A127374 A127375 A127376
%Y A127373 Sequence in context: A070677 A029584 A136255 this_sequence A050464 A014405 A143153
%K A127373 nonn,tabl
%O A127373 1,4
%A A127373 Gary W. Adamson (qntmpt(AT)yahoo.com), Jan 12 2007

%I A050464
%S A050464 0,0,1,0,0,2,1,0,3,0,1,4,0,2,6,0,0,6,1,0,10,2,1,8,0,0,10,4,0,12,
%T A050464 1,0,14,0,6,12,0,2,14,0,0,20,1,4,18,2,1,16,7,0,18,0,0,20,6,8,22,
%U A050464 0,1,24,0,2,31,0,0,28,1,0,26,12,1,24,0,0,31,4,18,28,1,0,30,0,1
%N A050464 Sum_{ d divides n, n/d=3 mod 4} d.
%Y A050464 Adjacent sequences: A050461 A050462 A050463 this_sequence A050465 A050466 A050467
%Y A050464 Sequence in context: A029584 A136255 A127373 this_sequence A014405 A143153 A127448
%K A050464 nonn
%O A050464 0,6
%A A050464 njas, Dec 23 1999

%I A014405
%S A014405 0,0,0,0,0,1,0,0,2,1,0,3,0,1,5,1,0,6,0,2,7,2,0,8,2,2,9,3,0,13,0,2,11,3,4,15,
%T A014405 0,3,13,6,0,18,0,4,20,4,0,19,2,8,18,5,0,23,6,6,20,5,0,30,0,5,25,6,7,29,0,6,
%U A014405 24,15,0,32,0,6,34,7,4,34,0,14,31,7,0,39,9,7,31,9,0,49,5,9,33,8,10,42,0,12
%N A014405 Number of arithmetic progressions of 3 or more positive integers, strictly increasing with sum n.
%e A014405 E.g. 15 = 1+2+3+4+5 = 1+5+9 = 2+5+8 = 3+5+7 = 4+5+6.
%o A014405 (PARI) a(n)= t=0; st=0; forstep(s=(n-3)\3,1,-1, st++; for(c=1,st, m=3; w=m*(s+c); while(w<n, w=w+s+m*c; m++); if(w==n, t++))); t - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Aug 30 2006
%Y A014405 Adjacent sequences: A014402 A014403 A014404 this_sequence A014406 A014407 A014408
%Y A014405 Sequence in context: A136255 A127373 A050464 this_sequence A143153 A127448 A128179
%K A014405 nonn
%O A014405 1,9
%A A014405 Clark Kimberling (ck6(AT)evansville.edu)

%I A143153
%S A143153 1,1,2,1,0,3,0,2,0,2,1,0,0,0,5,1,2,3,0,0,2,1,0,0,0,0,0,7,0,0,0,2,0,0,0,
%T A143153 2,0,0,3,0,0,0,0,0,3,1,2,0,0,5,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,11,0,2,0,2,
%U A143153 0,2,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,0,13,1,2,0,0,0,0,7,0,0,0,0,0,0,2
%V A143153 1,-1,2,-1,0,3,0,-2,0,2,-1,0,0,0,5,1,-2,-3,0,0,2,-1,0,0,0,0,0,7,0,0,0,-2,0,0,0,2,0,0,
%W A143153 -3,0,0,0,0,0,3,1,-2,0,0,-5,0,0,0,0,2,-1,0,0,0,0,0,0,0,0,0,11,0,2,0,-2,0,-2,0,0,0,0,0,
%X A143153 2,-1,0,0,0,0,0,0,0,0,0,0,0,13,1,-2,0,0,0,0,-7,0,0,0,0,0,0,2
%N A143153 Triangle read by rows, A054525 * (A020639 * 0^(n-k)), 1<=k<=n.
%C A143153 Right border = A020639, Lpf(n): (1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11,...).
%C A143153 Left border = mu(n), A008683: (1, -1, -1, 0, -1, 1, -1,...).
%C A143153 Row sums = a signed version of A097945: (1, -1, -2, 0, -4, 2, -6,...) such that parity = (+) iff mu(n) = +.
%F A143153 Triangle read by rows, A054525 * (A020639 * 0^(n-k)), 1<=k<=n; where A020639 = Lpf(n): (1, 2, 3, 2, 5, 2, 7, 2, 3, 2,...) and A054525 = the Mobius transform.
%e A143153 First few rows of the triangle =
%e A143153 1;
%e A143153 -1, 2;
%e A143153 -1, 0, 3;
%e A143153 0, -2, 0, 2;
%e A143153 -1, 0, 0, 0, 5;
%e A143153 1, -2, -3, 0, 0, 2;
%e A143153 -1, 0, 0, 0, 0, 0, 7;
%e A143153 0, 0, 0, -2, 0, 0, 0, 2;
%e A143153 0, 0, -3, 0, 0, 0, 0, 0, 3;
%e A143153 1, -2, 0, 0, -5, 0, 0, 0, 0, 2;
%e A143153 -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11;
%e A143153 0, 2, 0, -2, 0, -2, 0, 0, 0, 0, 0, 2;
%e A143153 ..
%Y A143153 Cf. A008683, A097945, A020639, A054525.
%Y A143153 Adjacent sequences: A143150 A143151 A143152 this_sequence A143154 A143155 A143156
%Y A143153 Sequence in context: A127373 A050464 A014405 this_sequence A127448 A128179 A058558
%K A143153 tabl,sign
%O A143153 1,3
%A A143153 Gary W. Adamson & Mats O. Granvik (qntmpkt(AT)yahoo.com), Jul 27 2008

%I A127448
%S A127448 1,1,2,1,0,3,0,2,0,4,1,0,0,0,5,1,2,3,0,0,6,1,0,0,0,0,0,7,0,0,0,4,0,0,0,
%T A127448 8,0,0,3,0,0,0,0,0,9,1,2,0,0,5,0,0,0,0,10
%V A127448 1,-1,2,-1,0,3,0,-2,0,4,-1,0,0,0,5,1,-2,-3,0,0,6,-1,0,0,0,0,0,7,0,0,0,-4,0,0,0,8,0,0,
%W A127448 -3,0,0,0,0,0,9,1,-2,0,0,5,0,0,0,0,10
%N A127448 Triangle, row sums = phi(n), left column = mu(n).
%C A127448 Left column = mu(n), A008683: (1, -1, 01, 0, -1, 1, -1,...). Row sums = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6, 4,...).
%F A127448 Mobius transform of matrix M, M = an infinite lower triangular matrix with (1, 2, 3,...) in the main diagonal and the rest zeros.
%e A127448 First few rows of the triangle are;
%e A127448 1;
%e A127448 -1, 2;
%e A127448 -1, 0, 3;
%e A127448 0, -2, 0, 4;
%e A127448 -1, 0, 0, 0, 5;
%e A127448 1, -2, -3, 0, 0, 6;
%e A127448 -1, 0, 0, 0, 0, 0, 7;
%e A127448 0, 0, 0, -4, 0, 0, 0, 8;
%e A127448 0, 0, -3, 0, 0, 0, 0, 0, 9;
%e A127448 1, -2, 0, 0, 5, 0, 0, 0, 0, 10;
%e A127448 ...
%Y A127448 Cf. A000010, A008683, A051731.
%Y A127448 Adjacent sequences: A127445 A127446 A127447 this_sequence A127449 A127450 A127451
%Y A127448 Sequence in context: A050464 A014405 A143153 this_sequence A128179 A058558 A123973
%K A127448 tabl,uned,sign
%O A127448 1,3
%A A127448 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 14 2007

%I A128179
%S A128179 1,0,2,1,0,3,0,2,0,4,1,0,3,0,5,0,2,0,4,0,6,1,0,3,0,5,0,7,0,2,0,4,0,6,0,
%T A128179 8,1,0,3,0,5,0,7,0,9,0,2,0,4,0,6,0,8,0,10
%N A128179 A097807 * A002260.
%C A128179 Row sums = A002620: (1, 2, 4, 6, 9, 12, 16, 20, 25, 30,...)
%F A128179 A097807 * A002260 as infinite lower triangular matrices. k-th column = (k, 0, k, 0,...).
%e A128179 First few rows of the triangle are:
%e A128179 1;
%e A128179 0, 2;
%e A128179 1, 0, 3;
%e A128179 0, 2, 0, 4;
%e A128179 1, 0, 3, 0, 5;
%e A128179 0, 2, 0, 4, 0, 6;
%e A128179 1, 0, 3, 0, 5, 0, 7;
%e A128179 ...
%Y A128179 Cf. A097807, A002260, A002620.
%Y A128179 Adjacent sequences: A128176 A128177 A128178 this_sequence A128180 A128181 A128182
%Y A128179 Sequence in context: A014405 A143153 A127448 this_sequence A058558 A123973 A098493
%K A128179 nonn,tabl
%O A128179 1,3
%A A128179 Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 17 2007

%I A058558
%S A058558 1,1,2,1,0,3,0,2,1,2,6,4,8
%V A058558 1,-1,-2,1,0,3,0,-2,1,-2,6,-4,-8
%N A058558 McKay-Thompson series of class 20c for Monster.
%D A058558 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%e A058558 T20c = 1/q - q - 2*q^3 + q^5 + 3*q^9 - 2*q^13 + q^15 - 2*q^17 + 6*q^19 - ...
%Y A058558 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%Y A058558 Adjacent sequences: A058555 A058556 A058557 this_sequence A058559 A058560 A058561
%Y A058558 Sequence in context: A143153 A127448 A128179 this_sequence A123973 A098493 A058560
%K A058558 sign
%O A058558 -1,3
%A A058558 njas, Nov 27, 2000

%I A123973
%S A123973 0,0,1,1,1,1,1,1,2,1,0,3,0,3,1,1,2,5,2,4,1,1,2,7,6,5,5,1,0,5,0,15,5,9,6,
%T A123973 1,1,3,12,9,25,1,14,7,1,1,3,15,18,29,35,7,20,8,1,0,7,0,42,14,63,42,20,
%U A123973 27,9,1
%V A123973 0,0,-1,-1,-1,1,-1,1,2,-1,0,3,0,-3,1,1,2,-5,-2,4,-1,1,-2,-7,6,5,-5,1,0,-5,0,15,-5,-9,6,
%W A123973 -1,-1,-3,12,9,-25,1,14,-7,1,-1,3,15,-18,-29,35,7,-20,8,-1,0,7,0,-42,14,63,-42,-20,27,
%X A123973 -9,1
%N A123973 Sequence of tridiagonal matrices with one center zero terminal that give a triangular sequence from the characteristic polynomials based on the 3 X 3 matrix type: {{1, -1, 0}, {-1, 1, -1}, {0, -1, 0}}.
%C A123973 Matrices: {{0}}, {{1, -1}, {-1, 0}}, {{1, -1, 0}, {-1, 1, -1}, {0, -1, 0}}, {{1, -1, 0, 0}, {-1, 1, -1, 0}, {0, -1, 1, -1}, {0, 0, -1, 0}}, {{1, -1, 0, 0, 0}, {-1, 1, -1, 0, 0}, {0, -1, 1, -1, 0}, {0, 0, -1, 1, -1}, {0, 0, 0, -1, 0}}, {{1, -1, 0, 0, 0, 0}, {-1, 1, -1, 0, 0, 0}, {0, -1, 1, -1, 0, 0}, {0, 0, -1, 1, -1, 0}, { 0, 0, 0, -1, 1, -1}, {0, 0, 0, 0, -1, 0}} Determinants ( not all Sl(3,Z) and invertable): Table[Det[M[d]], {d, 1, 10}] {0, -1, -1, 0, 1, 1, 0, -1, -1, 0}
%F A123973 t(n,m,d)=If[ n == m && n < d && m < d, 1, If[n == m - 1 || n == m + 1, -1, If[n == m == d, 0, 0]]]
%e A123973 Triangle begins:
%e A123973 {0},
%e A123973 {0, -1},
%e A123973 {-1, -1, 1},
%e A123973 {-1, 1, 2, -1},
%e A123973 {0, 3, 0, -3, 1},
%e A123973 {1, 2, -5, -2, 4, -1},
%e A123973 {1, -2, -7, 6, 5, -5, 1},
%e A123973 {0, -5, 0, 15, -5, -9, 6, -1},
%e A123973 {-1, -3, 12, 9, -25, 1, 14, -7, 1},
%e A123973 {-1, 3, 15, -18, -29, 35,7, -20, 8, -1},
%e A123973 {0, 7, 0, -42, 14, 63, -42, -20, 27, -9, 1}
%e A123973 Some of the polynomials are Steinbach.
%t A123973 T[n_, m_, d_] := If[ n == m && n < d && m < d, 1, If[n == m - 1 || n == m + 1, -1, If[n == m == d, 0, 0]]] M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] Table[M[d], {d, 1, 10}] Table[Det[M[d]], {d, 1, 10}] Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}] a = Join[M[1], Table[CoefficientList[Det[M[ d] - x*IdentityMatrix[d]], x], {d, 1, 10}]] Flatten[a] MatrixForm[a]
%Y A123973 Adjacent sequences: A123970 A123971 A123972 this_sequence A123974 A123975 A123976
%Y A123973 Sequence in context: A127448 A128179 A058558 this_sequence A098493 A058560 A131047
%K A123973 uned,probation,tabl,sign
%O A123973 1,9
%A A123973 Gary Adamson and Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 30 2006
%E A123973 Looking at the triangle suggests that the very first term should be 1, not 0. - njas, Nov 01, 2006

%I A098493
%S A098493 1,0,1,1,1,1,1,1,2,1,0,3,0,3,1,1,2,5,2,4,1,1,2,7,6,5,5,1,0,5,0,15,5,9,6,
%T A098493 1,1,3,12,9,25,1,14,7,1,1,3,15,18,29,35,7,20,8,1,0,7,0,42,14,63,42,20,
%U A098493 27,9,1,1,4,22,24,85,14,112,42
%V A098493 1,0,-1,-1,-1,1,-1,1,2,-1,0,3,0,-3,1,1,2,-5,-2,4,-1,1,-2,-7,6,5,-5,1,0,-5,0,15,-5,-9,6,
%W A098493 -1,-1,-3,12,9,-25,1,14,-7,1,-1,3,15,-18,-29,35,7,-20,8,-1,0,7,0,-42,14,63,-42,-20,27,
%X A098493 -9,1,1,4,-22,-24,85,14,-112,42
%N A098493 Triangle T(n,k) read by rows: difference between A098489 and A098490 at triangular rows.
%C A098493 Also, coefficients of polynomials that have values in A098495 and A094954.
%D A098493 A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, in preparation.
%F A098493 T(n, k) = A098489[n(n+1)/2, k] - A098490[n(n+1)/2, k].
%F A098493 Recurrence: T(n, k) = T(n-1, k)-T(n-1, k-1)-T(n-2, k); T(n, k)=0 for n<0, k>n, k<0; T(n, n)=(-1)^n; T(n, n-1)=(-1)^n*(1-n).
%e A098493 {1} {0,-1} {-1,-1,1} {-1,1,2,-1} {0,3,0,-3,1}...
%o A098493 (PARI) T(n,k)=if(k>n||k<0||n<0,0,if(k>=n-1,(-1)^n*if(k==n,1,-k),if(n==1,0,if(k==0,T(n-1,0)-T(n-2,0),T(n-1,k)-T(n-2,k)-T(n-1,k-1)))))
%Y A098493 Columns include A010892, -A076118. Diagonals include A033999, A038608, (-1)^n*A000096. Row sums are in A057077.
%Y A098493 Cf. A098494 (diagonal polynomials).
%Y A098493 Adjacent sequences: A098490 A098491 A098492 this_sequence A098494 A098495 A098496
%Y A098493 Sequence in context: A128179 A058558 A123973 this_sequence A058560 A131047 A143714
%K A098493 sign,tabl
%O A098493 0,9
%A A098493 Ralf Stephan (ralf(AT)ark.in-berlin.de), Sep 12 2004

%I A058560
%S A058560 1,2,1,0,3,0,3,4,0,2,3,8,4
%V A058560 1,2,-1,0,3,0,3,-4,0,2,3,8,-4
%N A058560 McKay-Thompson series of class 20e for Monster.
%D A058560 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%e A058560 T20e = 1/q + 2*q - q^3 + 3*q^7 + 3*q^11 - 4*q^13 + 2*q^17 + 3*q^19 + ...
%Y A058560 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%Y A058560 Adjacent sequences: A058557 A058558 A058559 this_sequence A058561 A058562 A058563
%Y A058560 Sequence in context: A058558 A123973 A098493 this_sequence A131047 A143714 A004172
%K A058560 sign
%O A058560 -1,2
%A A058560 njas, Nov 27, 2000

%I A131047
%S A131047 1,0,2,1,0,3,0,4,0,4,1,0,10,0,5,0,6,0,20,0,6,1,0,21,0,35,0,7,0,8,0,56,0,
%T A131047 56,0,8,1,0,36,0,126,0,84,0,9
%N A131047 (1/2) * ((A007318 - A007318^(-1)).
%C A131047 Row sums = (1, 2, 4, 8,...). A131047 * (1,2,3,...) = A087447 starting (1, 4, 10, 24, 56,...). A generalized set of analogous triangles: ((1/(Q+1)) * (P^Q - 1/P), Q an integer, generates triangles with row sums = powers of (Q+1). Cf. A131048, A131049, A131050, A131051 for triangles having Q = 2,3,4, and 5, respectively.
%C A131047 A007318, Pascal's triangle, = this triangle + A119467, since one triangle = the zeros or masks of the other. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 12 2007
%F A131047 Let A007318 (Pascal's triangle) = P, then A131047 = (1/2) * (P - 1/P); deleting the right border of zeros.
%e A131047 First few rows of the triangle are:
%e A131047 1;
%e A131047 0, 2;
%e A131047 1, 0, 3;
%e A131047 0, 4, 0, 4;
%e A131047 1, 0, 10, 0, 5;
%e A131047 0, 6, 0, 20, 0, 6;
%e A131047 1, 0, 21, 0, 35, 0, 7;
%e A131047 ...
%Y A131047 Cf. A131048, A131049, A131050, A131051.
%Y A131047 Cf. A119467.
%Y A131047 Adjacent sequences: A131044 A131045 A131046 this_sequence A131048 A131049 A131050
%Y A131047 Sequence in context: A123973 A098493 A058560 this_sequence A143714 A004172 A082754
%K A131047 nonn,tabl
%O A131047 1,3
%A A131047 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 12 2007

%I A143714
%S A143714 0,0,2,1,0,3,0,4,4,0,0,11,0,0,10,8,0,7,0,17,18,0,0,28,0,0,10,16,0,19,0,
%T A143714 15,18,0,6,33,0,0,14,42,0,35,0,16,42,0,0,77,0,0,18,19,0,19,24,53,20,0,0,
%U A143714 120,0,0,60,29,30,34,0,25,24,12,0,114,0,0,46,28,18,27,0,103,28,0,0,140
%N A143714 Number of pairs (a,b), 1 <= a <= b <= n, such that (a+b)^2+n^2 is a square.
%C A143714 Also: Number of cuboids of maximal side length n having integral shortest path going on the surface from one vertex to the opposite one.
%C A143714 Also: Number of subsets {a,b} of {1,..,n} such that (a+b,n) form the shorter two legs of a pythagorean triple.
%H A143714 <a href="http://projecteuler.net/index.php?section=problems&id=86">Project Euler: Problem 86</a>
%e A143714 For n=3, we have the 3 x 3 x 1 and the 3 x 2 x 2 cuboid, for which the shortest path on the surface from one vertex to the opposite is of integral length sqrt(3^2 + (2+2)^2) = sqrt(3^2 + (3+1)^2) = 5.
%e A143714 For n=4, there is the 4 x 2 x 1 cuboid having this property.
%e A143714 For n=1,2 and 5 there is no cuboid having this property, i.e. no s >= 2, s <= 2n such that s^2+n^2 would be a square.
%o A143714 (PARI) A143714(M)=sum(a=1,M,sum(b=a,M,issquare((a+b)^2+M^2)))
%Y A143714 Cf. A143715 (partial sums).
%Y A143714 Adjacent sequences: A143711 A143712 A143713 this_sequence A143715 A143716 A143717
%Y A143714 Sequence in context: A098493 A058560 A131047 this_sequence A004172 A082754 A063173
%K A143714 easy,nonn
%O A143714 1,3
%A A143714 M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Aug 29 2008

%I A004172
%S A004172 1,0,1,1,0,1,0,2,1,0,3,0,5,0,3,1,0,17,0,28,0,14,0,4,1,0,155,
%T A004172 0,255,0,126,0,30,0,5,1,0,2073,0,3410,0,1683,0,396,0,55,0,6,1,
%U A004172 0,38227,0,62881,0,31031,0,7293,0,1001,0,91,0,7,1,0,929569,0
%V A004172 1,0,-1,1,0,1,0,-2,1,0,-3,0,5,0,-3,1,0,17,0,-28,0,14,0,-4,1,0,-155,
%W A004172 0,255,0,-126,0,30,0,-5,1,0,2073,0,-3410,0,1683,0,-396,0,55,0,-6,1,
%X A004172 0,-38227,0,62881,0,-31031,0,7293,0,-1001,0,91,0,-7,1,0,929569,0
%N A004172 Triangle of coefficients of Euler polynomials E_2n(x) (exponents in increasing order).
%D A004172 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 809.
%H A004172 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b004172.txt">Rows n=0..50 of triangle, flattened</a>
%H A004172 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].
%H A004172 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerPolynomial.html">MathWorld: Euler Polynomial</a>
%Y A004172 Cf. A060083
%Y A004172 Adjacent sequences: A004169 A004170 A004171 this_sequence A004173 A004174 A004175
%Y A004172 Sequence in context: A058560 A131047 A143714 this_sequence A082754 A063173 A120111
%K A004172 sign,tabl,nice
%O A004172 0,8
%A A004172 njas

%I A082754
%S A082754 0,1,0,2,1,0,3,0,17,0,4,7,118,399,0,5,28,513,2800,7849,0,6,79,1844,
%T A082754 13983,61318,162287,0,7,192,6049,61440,357857,1417472,3667649,0,8,431,
%U A082754 18954,255583,1894076,9546255,35570638,91171007,0,9,924,58049,1038576
%N A082754 Triangle read by rows: T(n, k) = abs(n^k-k^n), 1<=k<=n.
%e A082754 0
%e A082754 1 0
%e A082754 2 1 0
%e A082754 3 0 17 0
%e A082754 4 7 118 399 0
%e A082754 5 28 513 2800 7849 0
%e A082754 ...
%Y A082754 Cf. A055651.
%Y A082754 Adjacent sequences: A082751 A082752 A082753 this_sequence A082755 A082756 A082757
%Y A082754 Sequence in context: A131047 A143714 A004172 this_sequence A063173 A120111 A130055
%K A082754 nonn,tabl
%O A082754 1,4
%A A082754 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 17 2003
%E A082754 More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Oct 04 2004

%I A063173
%S A063173 2,1,0,3,1,0,0,0,0,0,1,2,0,0,0,2,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,
%T A063173 0,4,1,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A063173 0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A063173 Prime-composite array T(m,n): highest power of the n-th prime that divides the n-th composite, read by antidiagonals.
%H A063173 N. Fernandez, <a href="http://www.borve.org/primeness/pcarray.html">The prime-composite array B(m,n) and the Borve conjectures</a>
%e A063173 Let p(n) be the n-th prime and c(m) the m-th composite. T(1,1)=2 because c(1)=4, p(1)=2 and the highest power of 2 in 4 is 2^2. T(1,2)=0 because c(1)=4, p(2)=3 and the highest power of 3 in 4 is 3^0. T(2,1)=1 because c(2)=6, p(1)=2 and the highest power of 2 in 6 is 2^1. So the sequence starts 2,0,1,...
%e A063173 Array begins
%e A063173 2 0 0 0 0 0 0 ...
%e A063173 1 1 0 0 0 0 0 ...
%e A063173 3 0 0 0 0 0 0 ...
%e A063173 0 2 0 0 0 0 0 ...
%e A063173 1 0 1 0 0 0 0 ...
%Y A063173 Cf. A000040, A002808, A063174, A063175, A063176.
%Y A063173 Adjacent sequences: A063170 A063171 A063172 this_sequence A063174 A063175 A063176
%Y A063173 Sequence in context: A143714 A004172 A082754 this_sequence A120111 A130055 A127013
%K A063173 nonn,tabl
%O A063173 1,1
%A A063173 N. Fernandez (primeness(AT)borve.org), Jul 09 2001

%I A120111
%S A120111 1,2,1,0,3,1,0,0,2,1,0,0,0,5,1,0,0,0,0,1,1,0,0,0,0,0,7,1,0,0,0,0,0,0,2,
%T A120111 1,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,11,1
%V A120111 1,-2,1,0,-3,1,0,0,-2,1,0,0,0,-5,1,0,0,0,0,-1,1,0,0,0,0,0,-7,1,0,0,0,0,0,0,-2,1,0,0,0,
%W A120111 0,0,0,0,-3,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,-11,1
%N A120111 Bi-diagonal inverse matrix of A120108.
%C A120111 Sub-diagonal is -lcm(1,...,n+2)/lcm(1,...,n+1) or -A014963(n+1). Row sums are A120112.
%e A120111 Triangle begins
%e A120111 1,
%e A120111 -2, 1,
%e A120111 0, -3, 1,
%e A120111 0, 0, -2, 1,
%e A120111 0, 0, 0, -5, 1,
%e A120111 0, 0, 0, 0, -1, 1,
%e A120111 0, 0, 0, 0, 0, -7, 1,
%e A120111 0, 0, 0, 0, 0, 0, -2, 1,
%e A120111 0, 0, 0, 0, 0, 0, 0, -3, 1,
%e A120111 0, 0, 0, 0, 0, 0, 0, 0, -1, 1,
%e A120111 0, 0, 0, 0, 0, 0, 0, 0, 0, -11, 1
%Y A120111 Adjacent sequences: A120108 A120109 A120110 this_sequence A120112 A120113 A120114
%Y A120111 Sequence in context: A004172 A082754 A063173 this_sequence A130055 A127013 A117362
%K A120111 easy,sign,tabl
%O A120111 0,2
%A A120111 Paul Barry (pbarry(AT)wit.ie), Jun 09 2006

%I A130055
%S A130055 1,0,2,1,0,3,1,0,0,4,3,0,0,0,5,0,2,0,0,0,6,5,0,0,0,0,0,7,2,2,0,0,0,0,0,
%T A130055 8,3,0,3,0,0,0,0,0,9,0,6,0,0,0,0,0,0,0,10
%V A130055 1,0,2,-1,0,3,-1,0,0,4,-3,0,0,0,5,0,-2,0,0,0,6,-5,0,0,0,0,0,7,-2,-2,0,0,0,0,0,8,-3,0,
%W A130055 -3,0,0,0,0,0,9,0,-6,0,0,0,0,0,0,0,10
%N A130055 A129691 * A127093.
%C A130055 Row sums = d(n), A000005: (1, 2, 2, 3, 2, 4, 2, 4, 3, 4,...). Left column = A130054: (1, 0, -1, -1, -3, 0, -5, -2, -3, 0,...).
%F A130055 A129691 * A127093 as infinite lower triangular matrices.
%e A130055 First few rows of the triangle are:
%e A130055 1;
%e A130055 0, 2;
%e A130055 -1, 0, 3;
%e A130055 -1, 0, 0, 4;
%e A130055 -3, 0, 0, 0, 5;
%e A130055 0, -2, 0, 0, 0, 6;
%e A130055 -5, 0, 0, 0, 0, 0, 7;
%e A130055 ...
%Y A130055 Cf. A129691, A127093, A000005, A130054.
%Y A130055 Adjacent sequences: A130052 A130053 A130054 this_sequence A130056 A130057 A130058
%Y A130055 Sequence in context: A082754 A063173 A120111 this_sequence A127013 A117362 A113214
%K A130055 tabl,sign
%O A130055 1,3
%A A130055 Gary W. Adamson (qntmpkt(AT)yahoo.com), May 04 2007

%I A127013
%S A127013 1,1,2,1,0,3,1,0,2,4,1,0,0,0,5,1,0,0,2,3,6,1,0,0,0,0,0,7,1,0,0,0,2,0,4,
%T A127013 8,1,0,0,0,0,0,3,0,9,1,0,0,0,0,2,0,0,5,1,0
%N A127013 Triangle read by rows: reversal of A126988.
%C A127013 Let j = reversed indices of row terms. Then for any row, j*T(n,k) = n, for non-zero T(n,k). For example, in row 10, we match the terms with their j indices: (1, 0, 0, 0, 0, 2, 0, 0, 5, 10), (dot product) (10, 9, 8, 7, 6, 5, 4, 3, 2, 1); getting 10, 0, 0, 0, 0, 10, 0, 0, 10, 10). The factors of n are found in each row in order, as non-zero terms; e.g. 10 has the factors 1, 2, 5, 10, sum 18. Row sums = sigma(n), A000203.
%D A127013 David Wells, "Prime Numbers, The Most Mysterious Figures in Math", John Wiley & Sons, 2005, Appendix.
%F A127013 Reversed rows of A126988
%e A127013 Row 10 = (1, 0, 0, 0, 0, 2, 0, 0, 5, 10), reversal of 10-th row of A126988.
%Y A127013 Cf. A126988, A000203.
%Y A127013 Adjacent sequences: A127010 A127011 A127012 this_sequence A127014 A127015 A127016
%Y A127013 Sequence in context: A063173 A120111 A130055 this_sequence A117362 A113214 A029323
%K A127013 nonn,tabl
%O A127013 1,3
%A A127013 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 02 2007

%I A117362
%S A117362 1,2,1,0,3,1,0,2,4,1,0,0,5,5,1,0,0,2,9,6,1,0,0,0,7,14,7,1,0,0,0,2,16,20,
%T A117362 8,1,0,0,0,0,9,30,27,9,1,0,0,0,0,2,25,50,35,10,1,0,0,0,0,0,11,55,77,44,
%U A117362 11,1
%V A117362 1,-2,1,0,-3,1,0,2,-4,1,0,0,5,-5,1,0,0,-2,9,-6,1,0,0,0,-7,14,-7,1,0,0,0,2,-16,20,-8,1,
%W A117362 0,0,0,0,9,-30,27,-9,1,0,0,0,0,-2,25,-50,35,-10,1,0,0,0,0,0,-11,55,-77,44,-11,1
%N A117362 Riordan array (1-2x,x(1-x)).
%C A117362 A signed version of A113214. Inverse of A092392. Row sums are A100051(n+1) with g.f. (1-2x)/(1-x+x^2). Diagonal sums are A117363.
%F A117362 Number triangle T(n,k)=(-1)^(n-k)(C(k,n-k)+2*C(k, n-k-1))
%e A117362 Triangle begins
%e A117362 1,
%e A117362 -2, 1,
%e A117362 0, -3, 1,
%e A117362 0, 2, -4, 1,
%e A117362 0, 0, 5, -5, 1,
%e A117362 0, 0, -2, 9, -6, 1,
%e A117362 0, 0, 0, -7, 14, -7, 1,
%e A117362 0, 0, 0, 2, -16, 20, -8, 1
%Y A117362 Adjacent sequences: A117359 A117360 A117361 this_sequence A117363 A117364 A117365
%Y A117362 Sequence in context: A120111 A130055 A127013 this_sequence A113214 A029323 A071802
%K A117362 easy,sign,tabl
%O A117362 0,2
%A A117362 Paul Barry (pbarry(AT)wit.ie), Mar 10 2006

%I A113214
%S A113214 1,2,1,0,3,1,0,2,4,1,0,0,5,5,1,0,0,2,9,6,1,0,0,0,7,14,7,1,0,0,0,2,16,20,
%T A113214 8,1,0,0,0,0,9,30,27,9,1,0,0,0,0,2,25,50,35,10,1,0,0,0,0,0,11,55,77,44,
%U A113214 11,1,0,0,0,0,0,2,36,105,112,54,12,1,0,0,0,0,0,0,13,91,182,156,65,13,1
%N A113214 Riordan array (1+2x,x(1+x)).
%C A113214 Row sums are Lucas numbers A000204. Diagonal sums are A007307(n+1). Inverse is (-1)^(n-k)A092392(n,k). Product with Pascal triangle (1/(1-x),x/(1-x)) is A111125.
%F A113214 T(n, k)=C(k, n-k)+2C(k, n-k-1); T(n, k)=sum{j=0..n, (-1)^(n-j)*C(n, j)C(j+k, 2k)(2j+1)/(2k+1)}.
%e A113214 Triangle begins
%e A113214 1;
%e A113214 2,1;
%e A113214 0,3,1;
%e A113214 0,2,4,1;
%e A113214 0,0,5,5,1;
%e A113214 0,0,2,9,6,1;
%e A113214 0,0,0,7,14,7,1;
%e A113214 0,0,0,2,16,20,8,1;
%Y A113214 Adjacent sequences: A113211 A113212 A113213 this_sequence A113215 A113216 A113217
%Y A113214 Sequence in context: A130055 A127013 A117362 this_sequence A029323 A071802 A110355
%K A113214 easy,nonn,tabl
%O A113214 0,2
%A A113214 Paul Barry (pbarry(AT)wit.ie), Oct 18 2005

%I A029323
%S A029323 1,0,0,1,0,0,1,0,0,2,1,0,3,1,0,3,1,0,4,2,1,5,3,1,6,3,1,
%T A029323 7,4,2,9,5,3,10,6,3,12,7,4,14,9,5,16,10,6,18,12,7,21,14,
%U A029323 9,23,16,10,26,18,12,29,21,14,33,23,16,36,26,18,40,29,21
%N A029323 Expansion of 1/((1-x^3)(1-x^9)(1-x^10)(1-x^12)).
%Y A029323 Adjacent sequences: A029320 A029321 A029322 this_sequence A029324 A029325 A029326
%Y A029323 Sequence in context: A127013 A117362 A113214 this_sequence A071802 A110355 A029293
%K A029323 nonn
%O A029323 0,10
%A A029323 njas

%I A071802
%S A071802 1,0,2,1,0,3,1,0,4,1,0,5,2,0,6,1,0,7,1,0,8,4,3,1,0,9,1,0,10,3,0,11,2,0,
%T A071802 12,3,0,13,4,3,1,0,14,5,0,15,1,0,16,5,3,1,0,17,3,0,18,3,0,19,5,2,1,0,
%U A071802 20,3,0,21,2,0,22,1,0,23,5,0,24,4,3,1,0,25,3,0,26,4,3,1,0,27,5,2,1,0
%N A071802 Table in which n-th row gives exponents (in decreasing order) of lexicographically earliest primitive irreducible polynomial of degree n over GF(2).
%D A071802 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1978, p. 408.
%D A071802 M. Olofsson, VLSI Aspects on Inversion in Finite Fields, Dissertation No. 731, Dept Elect. Engin., Linkoping, Sweden, 2002.
%e A071802 x+1, x^2+x+1, x^3+x+1, x^4+x+1, x^5+x^2+1, ...
%t A071802 a = {}; Do[k = 2^n + 1; While[s = Apply[Plus, IntegerDigits[k, 2]*x^Table[i, {i, n, 0, -1}]]; k < 2^(n + 1) - 1 && Factor[s, Modulus -> 2] =!= s, k += 2]; a = Append[a, Reverse[ Exponent[ Apply[ Plus, IntegerDigits[k, 2]*x^Table[i, {i, n, 0, -1}]], x, List]]], {n, 1, 27}]; Flatten[a]
%Y A071802 Cf. A058943.
%Y A071802 Adjacent sequences: A071799 A071800 A071801 this_sequence A071803 A071804 A071805
%Y A071802 Sequence in context: A117362 A113214 A029323 this_sequence A110355 A029293 A029312
%K A071802 nonn,nice,easy,tabf
%O A071802 1,3
%A A071802 njas, Jun 24 2002
%E A071802 Extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 25 2002

%I A110355
%S A110355 1,0,2,1,0,3,1,0,4,2,0,5,2,0,6,3,1,0,7,3,1,0,8,4,1,0,9,4,1,0,10,5,1,0,
%T A110355 11,5,1,0,12,6,2,0,13,6,2,0,14,7,2,0,15,7,2,0,16,8,2,0,17,8,2,0,18,9,3,
%U A110355 0,19,9,3,0,20,10,3,0,21,10,3,0,22,11,3,0,23,11,3,0,24,12,4,1,0,25,12,4
%N A110355 Following array contains numbers INT[n/1], INT[INT[n/1]/2],... until a zero is obtained. Sequence contains the array by rows.
%C A110355 1 0
%C A110355 2 1 0
%C A110355 3 1 0
%C A110355 4 2 0
%C A110355 5 2 0
%C A110355 6 3 1 0
%C A110355 ...
%Y A110355 Adjacent sequences: A110352 A110353 A110354 this_sequenc