The On-Line Encyclopedia of Integer Sequences, Recent Additions This is a section of the main database for the On-Line Encyclopedia of Integer Sequences. It shows the most recently added sequences in reverse chronological order. For more information see the following pages: ( www.research.att.com/~njas/sequences/ then ) Seis.html: Welcome index.html: Lookup demo1.html: Demos Sindx.html: Index WebCam.html: WebCam Submit.html: Contribute new sequence or comment eishelp1.html: Internal format eishelp2.html: Beautified format transforms.html: Transforms Spuzzle.html: Puzzles Shot.html: Hot classic.html: Classics ol.html: Superseeker pages.html: More pages Maintained by: N. J. A. Sloane (njas@research.att.com), home page: www.research.att.com/~njas/ The WebCam at www.research.att.com/~njas/sequences/WebCam.html is another way to browse the recent additions. [If the database has just been resorted into lexicographic order, the present file will be empty, but the WebCam will still work.] (start) %I A143665 %S A143665 2,3,31,13,11,7,19531,313,19,521,12207031,601,305175781,29,181,17,409, %T A143665 5167,191,41,379,23 %N A143665 a(n) is the least prime such that the multiplicative order of 5 mod a(n) equals to n. %Y A143665 Cf. A064081 A143663 A112092. %K A143665 nonn,new %O A143665 1,1 %A A143665 Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 28 2008 %I A112092 %S A112092 3,5,7,17,11,13,43,257,19,41,23,241,2731,29,151,65537,43691,37,174763, %T A112092 61681,337,397,47,97,251,53,87211,15790321 %N A112092 a(n) is the least prime such that the multiplicative order of 4 mod a(n) equals to n. %C A112092 a(n) is the minimal prime divisor of A064080(n). %Y A112092 Cf. A064080 A143663. %K A112092 nonn,new %O A112092 1,1 %A A112092 Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 28 2008 %I A143663 %S A143663 2,1,13,5,11,7,1093,41,757,61,23,73,797161,547,4561,17,1871,19,1597, %T A143663 1181,368089,67,47,6481,8951,398581,109,29,59,31,683,21523361 %N A143663 a(n) is the least prime such that the multiplicative order of 3 mod a(n) equals to n, and a(n)=1 if such prime does not exist. %C A143663 If a(n) differs from 1, then a(n) is the minimal prime divisor of A064079(n). %Y A143663 Cf. A002326 A064079. %K A143663 nonn,new %O A143663 1,1 %A A143663 Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 28 2008 %I A143656 %S A143656 1,1,0,1,1,0,1,0,2,0,1,1,2,3,0,1,0,0,0,7,0,1,1,2,3,7,8,0,1,0,2,0,7,0,22, %T A143656 0,1,1,0,3,7,0,22,32,0,1,0,2,0,0,0,22,0,66,0,1,1,2,3,7,8,22,32,66,91,0, %U A143656 1,0,0,0,7,0,22,0,00,233,0,1,1,2,3,7,8,22,32,66,91,233,263,1,0,2,0,7,0 %N A143656 Triangle read by rows, A054521 * (A045545 * 0^(n-k)); 1<=k<=n. %C A143656 Sum of row terms = A045545 starting with offset 1: (1, 1, 2, 3, 7, 8, 22,...). %C A143656 A045545 also = rightmost diagonal with nonzero terms. %C A143656 Sum of n-th row terms = rightmost nonzero term of next row. %C A143656 Prime n rows = first (n-1) terms of (1, 1, 2, 3, 7, 8,...) followed by 0. %C A143656 Asymptotic limit of A054521^n * A143656 = A045545 as a vector. %F A143656 Triangle read by rows, A054521 * (A045545 * 0^(n-k)); 1<=k<=n. T(n,k) = A045545(k) if gcd(n,k) = 1, 0 otherwise; where A045545 = (1, 1, 2, 3, 7, 8, 22, 32, 66,...) starting with offset 1. %e A143656 First few rows of the triangle = %e A143656 1; %e A143656 1, 0; %e A143656 1, 1, 0; %e A143656 1, 0, 2, 0; %e A143656 1, 1, 2, 3, 0; %e A143656 1, 0, 0, 0, 7, 0; %e A143656 1, 1, 2, 3, 7, 8, 0; %e A143656 1, 0, 2, 0, 7, 0, 22, 0; %e A143656 1, 1, 0, 3, 7, 0, 22, 32, 0; %e A143656 1, 0, 2, 0, 0, 0, 22, 0, 66, 0; %e A143656 .. %Y A143656 Cf. A054521, A045545. %K A143656 nonn,tabl,new %O A143656 1,9 %A A143656 Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 28 2008 %I A143655 %S A143655 0,0,0,0,2,0,0,0,3,0,0,2,3,0,0,0,0,0,0,5,0,0,2,3,0,5,0,0,0,0,3,0,5,0,7, %T A143655 0,0,2,0,0,5,0,7,0,0,0,0,3,0,0,0,7,0,0,0,0,2,3,0,5,0,7,0,0,0,0,0,0,0,0, %U A143655 5,0,7,0,0,0,11,0,0,2,3,0,5,0,7,0,0,0,11,0,0,0,0,3,0,5,0,0,0,0,0,11,0 %N A143655 Triangle read by rows, primes not dividing n; A054521 * (A061397 * 0^(n-k)), 1<=k<=n. %C A143655 Row sums = A066911: (0, 0, 2, 3, 5, 5, 10, 15, 14,....) %F A143655 Triangle read by rows, A054521 * (A061397 * 0^(n-k)), 1<=k<=n. T(n,k) = prime if k is prime but does not divide n. A054521 = a triangle with row sums phi(n). A061397 = (0, 2, 3, 0, 5, 0, 7,...) %e A143655 First few rows of the triangle = %e A143655 0; %e A143655 0, 0; %e A143655 0, 2, 0; %e A143655 0, 0, 3, 0; %e A143655 0, 2, 3, 0, 0; %e A143655 0, 0, 0, 0, 5, 0; %e A143655 0, 2, 3, 0, 5, 0, 0; %e A143655 0, 0, 3, 0, 5, 0, 7, 0; %e A143655 .. %e A143655 Row 8 has 3 primes < 8 not dividing 8: (3, 5, 7); where (3 + 5 + 7) = A066911(8). %Y A143655 Cf. A061397, A066911, A054521. %K A143655 nonn,tabl,new %O A143655 1,5 %A A143655 Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 28 2008 %I A143597 %S A143597 1,1,1,3,19,297,8953,572155,72116459,18460128753,9414877745601, %T A143597 9640779710687955,19725063387945457219,80793830752052788593529, %U A143597 661701532957780822275151305,10841317673677535233876159099755 %N A143597 G.f. satisfies: A(x) = 1 + x*A(2x)*A(-x). %F A143597 G.f. satisfies: A(x) = (1 + x*A(2x))/(1 + x^2*A(2x)*A(-2x)). %F A143597 a(n) = Sum_{k=0..n-1} 2^k*(-1)^(n-1-k)*a(k)*a(n-1-k) for n>0 with a(0)=1. %e A143597 G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 19*x^4 + 297*x^5 + 8953*x^6 +... %e A143597 A(x) = 1 + x*A(2x)*[1 - x*A(-2x)*[1 + x*A(2x)*[1 - x*A(-2x)*[1 +...]]]]. %o A143597 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,2*x)*subst(A,x,-x));polcoeff(A,n)} %o A143597 (PARI) {a(n)=if(n==0,1,sum(k=0,n-1,2^k*(-1)^(n-1-k)*a(k)*a(n-1-k)))} %K A143597 nonn,new %O A143597 0,4 %A A143597 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 28 2008 %I A143617 %S A143617 0,8,10,18,20,28,68,88,108,188,200,208,288,688,888,1088,1888,2008,2088, %T A143617 2888,6888,8888,10888,18888,20088,20888,28888,68888,88888,108888,188888, %U A143617 200888,208888,288888,688888,888888,1088888,1888888,2008888,2088888 %N A143617 Where record values occur in A010371. %C A143617 a(n+7) = 10*a(n) + 8 for n > 4; %C A143617 A010371(a(n))=A143616(n) and A010371(m)Project Euler, Problem 200. %o A143611 (PARI) /* return 1 if no digit can be changed to make it prime; if d=1, print a prime if n is not prime-proof */ isA143611(n,d=0)={ forstep( k=n\10*10+1,n\10*10+9,2, isprime(k) | next; d & print("prime:",k); return); n%2 | return(1);/* even : no other digit can make it prime */ n%5 | return(1);/* ending in 0 or 5: idem */ o=10; until( n < o*=10, k=n-o*(n\o%10); for( i=0,9, isprime(k) & return(d & print("prime:",k)); k+=o));1} %Y A143611 Cf. A143641. %K A143611 base,easy,nonn,new %O A143611 1,1 %A A143611 M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Aug 27 2008 %I A143643 %S A143643 1,3,5,12,19,45,71,168,265,627,989,2340,3691,8733,13775,32592,51409, %T A143643 121635,191861,453948,716035,1694157,2672279,6322680,9973081,23596563 %N A143643 Lower principal and intermediate convergents to 3^(1/2). %C A143643 The lower principal and intermediate convergents to 3^(1/2), beginning with %C A143643 1/1, 3/2, 5/3, 12/7, 19/11, form a strictly increasing sequence; essentially, numerators=A143643 and denominators=A005246. %D A143643 Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966. %D A143643 Clark Kimberling, "Best lower and upper approximates to irrational numbers," Elemente der Mathematik, 52 (1997) 122-126. %K A143643 nonn,new %O A143643 1,2 %A A143643 Clark Kimberling (ck6(AT)evansville.edu), Aug 27 2008 %I A143642 %S A143642 1,2,3,5,7,12,19,26,45,71,97,168,265,362,627,989,1351,2340,3691,5042, %T A143642 8733,13775,18817,32592,51409,70226,121635,191861,262087,453948,716035, %U A143642 978122,1694157,2672279,3650401,6322680,9973081,13623482,23596563 %N A143642 Numerators of principal and intermediate convergents to 3^(1/2). %C A143642 The first few principal and intermediate convergents to 3^(1/2) are %C A143642 1/1, 2/1, 3/2, 5/3, 7/4, 12/7; essentially, numerators=A143642 and %C A143642 denominators=A140827. %D A143642 Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966. %D A143642 Clark Kimberling, "Best lower and upper approximates to irrational numbers," Elemente der Mathematik, 52 (1997) 122-126. %K A143642 nonn,new %O A143642 1,2 %A A143642 Clark Kimberling (ck6(AT)evansville.edu), Aug 27 2008 %I A143641 %S A143641 212159,595631,872897,1203623,1293671,1566691,1702357,1830661,3716213, %T A143641 3964169,4103917,4134953,4173921,4310617,4376703,4586509,4703801, %U A143641 4749187,4801387,4928909,5005353,5051179,5231739,5258901,5317573 %N A143641 Odd prime-proof numbers (A143611) not ending in 5. %C A143641 Most "prime-proof" numbers are even or multiple of 5, cf. A143611. %H A143641 Project Euler, Problem 200. %o A143641 (PARI) forstep( i=1,10^7,2, i%5 || next; isA143611(i) && print1(i",")) %Y A143641 Cf. A143611. %K A143641 base,nonn,new %O A143641 1,1 %A A143641 M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Aug 27 2008 %I A143609 %S A143609 2,3,10,17,58,99,338,577,1970,3363,11482,19601,66922,114243,390050, %T A143609 665857,2273378,3880899,13250218,22619537,77227930 %N A143609 Numerators of the upper principal and intermediate convergents to 2^(1/2). %C A143609 The upper principal and intermediate convergents to 2^(1/2), beginning with %C A143609 2/1, 3/2, 10/7, 17/12, 58/41, form a strictly decreasing sequence; %C A143609 essentially, numerators=A143609 and denominators=A084068. %D A143609 Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966. %D A143609 Clark Kimberling, "Best lower and upper approximates to irrational numbers," Elemente der Mathematik, 52 (1997) 122-126. %K A143609 nonn,new %O A143609 1,1 %A A143609 Clark Kimberling (ck6(AT)evansville.edu), Aug 27 2008 %I A143608 %S A143608 1,4,7,24,41,140,239,816,1393,4756,8119,27720,47321,161564,275807, %T A143608 941664,1607521,5488420,9369319,31988856,54608393 %N A143608 Numerators of the lower principal and intermediate convergents to 2^(1/2). %C A143608 The lower principal and intermediate convergents to 2^(1/2), beginning with %C A143608 1/1, 4/3, 7/5, 24/17, 41/29, form a strictly increasing sequence; %C A143608 essentially, numerators=A143608 and denominators=A079496. %D A143608 Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966. %D A143608 Clark Kimberling, "Best lower and upper approximates to irrational numbers," Elemente der Mathematik, 52 (1997) 122-126. %K A143608 nonn,new %O A143608 1,2 %A A143608 Clark Kimberling (ck6(AT)evansville.edu), Aug 27 2008 %I A143607 %S A143607 1,3,4,7,10,17,24,41,58,99,140,239,338,577,816,1393,1970,3363,4756,8119, %T A143607 11482,19601,27720,47321,66922,114243,161564,275807,390050,665857, %U A143607 941664,1607521,2273378,3880899,5488420,9369319,13250218,22619537 %N A143607 Numerators of principal and intermediate convergents to 2^(1/2). %C A143607 The principal and intermediate convergents to 2^(1/2) begin with 1/1, 3/2 4/3, 7/5, 10/7; essentially, numerators=A143607, denominators=A002965. %D A143607 Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966. %D A143607 Clark Kimberling, "Best lower and upper approximates to irrational numbers," Elemente der Mathematik, 52 (1997) 122-126. %K A143607 nonn,new %O A143607 1,2 %A A143607 Clark Kimberling (ck6(AT)evansville.edu), Aug 27 2008 %I A143640 %S A143640 1,1,3,40,829,26096,1216327,76192824,6123167801,615764308672, %T A143640 75666884850091,11126407433017944,1925795142055097557, %U A143640 387184416676122044032,89407267196505737775311 %N A143640 E.g.f. satisfies: A(x) = exp(x*A(((x+1)^9-1)/9)). %p A143640 A:= proc(n,k::nonnegint) option remember; if n<=0 or k=0 then 1 else A(n-1,k)(((x+1)^k-1)/k) fi; unapply (convert (series (exp (x*%), x,n+1), polynom), x) end: a:= n-> coeff (A(n,9)(x), x,n)*n!: seq (a(n), n=0..20); %Y A143640 Cf. 9th column of A143632. %K A143640 nonn,new %O A143640 0,3 %A A143640 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 27 2008 %I A143639 %S A143639 1,1,3,37,713,20931,900067,51768739,3815631297,351259985449, %T A143639 39429531406511,5287999813256799,833815716731955817, %U A143639 152569133029591977895,32033950906843181020467 %N A143639 E.g.f. satisfies: A(x) = exp(x*A(((x+1)^8-1)/8)). %p A143639 A:= proc(n,k::nonnegint) option remember; if n<=0 or k=0 then 1 else A(n-1,k)(((x+1)^k-1)/k) fi; unapply (convert (series (exp (x*%), x,n+1), polynom), x) end: a:= n-> coeff (A(n,8)(x), x,n)*n!: seq (a(n), n=0..20); %Y A143639 Cf. 8th column of A143632. %K A143639 nonn,new %O A143639 0,3 %A A143639 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 27 2008 %I A143638 %S A143638 1,1,3,34,605,16416,644647,33690574,2252245353,187575203080, %T A143638 19000833293771,2295318297423834,325536649109809117, %U A143638 53508774130762119508,10080999100649218887615,2156137639664134179951166 %N A143638 E.g.f. satisfies: A(x) = exp(x*A(((x+1)^7-1)/7)). %p A143638 A:= proc(n,k::nonnegint) option remember; if n<=0 or k=0 then 1 else A(n-1,k)(((x+1)^k-1)/k) fi; unapply (convert (series (exp (x*%), x,n+1), polynom), x) end: a:= n-> coeff (A(n,7)(x), x,n)*n!: seq (a(n), n=0..20); %Y A143638 Cf. 7th column of A143632. %K A143638 nonn,new %O A143638 0,3 %A A143638 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 27 2008 %I A143637 %S A143637 1,1,3,31,505,12521,443227,20766159,1240975409,92068494625, %T A143637 8282460205891,886498379552919,111190541933344777,16136424098890466281, %U A143637 2680205744964849259355,504746978220729054647911 %N A143637 E.g.f. satisfies: A(x) = exp(x*A(((x+1)^6-1)/6)). %p A143637 A:= proc(n,k::nonnegint) option remember; if n<=0 or k=0 then 1 else A(n-1,k)(((x+1)^k-1)/k) fi; unapply (convert (series (exp (x*%), x,n+1), polynom), x) end: a:= n-> coeff (A(n,6)(x), x,n)*n!: seq (a(n), n=0..20); %Y A143637 Cf. 6th column of A143632. %K A143637 nonn,new %O A143637 0,3 %A A143637 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 27 2008 %I A143636 %S A143636 1,1,3,28,413,9216,289111,11925964,624637785,40422282112,3159287760491, %T A143636 292875271947468,31733363437993285,3969285168539789008, %U A143636 567118401777735330447,91714059231986721233596 %N A143636 E.g.f. satisfies: A(x) = exp(x*A(((x+1)^5-1)/5)). %p A143636 A:= proc(n,k::nonnegint) option remember; if n<=0 or k=0 then 1 else A(n-1,k)(((x+1)^k-1)/k) fi; unapply (convert (series (exp (x*%), x,n+1), polynom), x) end: a:= n-> coeff (A(n,5)(x), x,n)*n!: seq (a(n), n=0..21); %Y A143636 Cf. 5th column of A143632. %K A143636 nonn,new %O A143636 0,3 %A A143636 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 27 2008 %I A143635 %S A143635 1,1,3,25,329,6471,175747,6222259,277683681,15206462497,1000136567591, %T A143635 77666331244239,7021789807671817,730394622232111747, %U A143635 86529393614846902371,11573498785704862459891,1734360074041552070631713 %N A143635 E.g.f. satisfies: A(x) = exp(x*A(((x+1)^4-1)/4)). %p A143635 A:= proc(n,k::nonnegint) option remember; if n<=0 or k=0 then 1 else A(n-1,k)(((x+1)^k-1)/k) fi; unapply (convert (series (exp (x*%), x,n+1), polynom), x) end: a:= n-> coeff (A(n,4)(x), x,n)*n!: seq (a(n), n=0..21); %Y A143635 Cf. 4th column of A143632. %K A143635 nonn,new %O A143635 0,3 %A A143635 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 27 2008 %I A143634 %S A143634 1,1,3,22,253,4256,96727,2828274,102988937,4553158024,239618067211, %T A143634 14775790894734,1053758625896077,85965003368491300,7947211237328151167, %U A143634 825821792546485330306,95772123012223308982673 %N A143634 E.g.f. satisfies: A(x) = exp(x*A(((x+1)^3-1)/3)). %p A143634 A:= proc(n,k::nonnegint) option remember; if n<=0 or k=0 then 1 else A(n-1,k)(((x+1)^k-1)/k) fi; unapply (convert (series (exp (x*%), x,n+1), polynom), x) end: a:= n-> coeff (A(n,3)(x), x,n)*n!: seq (a(n), n=0..21); %Y A143634 Cf. 3rd column of A143632. %K A143634 nonn,new %O A143634 0,3 %A A143634 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 27 2008 %I A143633 %S A143633 1,1,3,19,185,2541,45787,1037359,28649553,942585625,36294146171, %T A143633 1612599520599,81729515092777,4679679856932133,300257015404355115, %U A143633 21436580394615666991,1692530428442960006753,146987828523665177048241 %N A143633 E.g.f. satisfies: A(x) = exp(x*A(((x+1)^2-1)/2)). %p A143633 A:= proc(n,k::nonnegint) option remember; if n<=0 or k=0 then 1 else A(n-1,k)(((x+1)^k-1)/k) fi; unapply (convert (series (exp (x*%), x,n+1), polynom), x) end: a:= n-> coeff (A(n,2)(x), x,n)*n!: seq (a(n), n=0..21); %Y A143633 Cf. 2nd column of A143632. %K A143633 nonn,new %O A143633 0,3 %A A143633 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 27 2008 %I A143632 %S A143632 1,1,1,1,1,1,1,1,3,1,1,1,3,16,1,1,1,3,19,125,1,1,1,3,22,185,1296,1,1,1, %T A143632 3,25,253,2541,16807,1,1,1,3,28,329,4256,45787,262144,1,1,1,3,31,413, %U A143632 6471,96727,1037359,4782969,1,1,1,3,34,505,9216,175747,2828274,28649553 %N A143632 Table T(n,k), n>=0, k>=0, read by antidiagonals, where the e.g.f. for column k satisfies A_k(x) = exp(x*A_k(((x+1)^k-1)/k)) if k>0 and A_0(x) = exp(x*A_0(0)) = exp(x). %e A143632 Table begins: %e A143632 1, 1, 1, 1, 1, 1 %e A143632 1, 1, 1, 1, 1, 1 %e A143632 1, 3, 3, 3, 3, 3 %e A143632 1, 16, 19, 22, 25, 28 %e A143632 1, 125, 185, 253, 329, 413 %e A143632 1, 1296, 2541, 4256, 6471, 9216 %p A143632 A:= proc(n,k::nonnegint) option remember; if n<=0 or k=0 then 1 else A(n-1,k)(((x+1)^k-1)/k) fi; unapply (convert (series (exp (x*%), x,n+1), polynom), x) end: T:= (n,k)-> coeff (A(n,k)(x), x,n)*n!: seq (seq(T(n,d-n), n=0..d), d=0..11); %Y A143632 Cf. columns 0-9: A000012, A000272, A143633, A143634, A143635, A143636, A143637, A143638, A143639, A143640. %K A143632 nonn,tabl,new %O A143632 0,9 %A A143632 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 27 2008 %I A143620 %S A143620 1,1,0,2,0,0,2,1,0,0,4,1,1,0,0,2,1,1,1,0,0,6,2,2,1,1,0,0,4,3,2,2,1,1,0, %T A143620 0,6,2,4,2,2,1,1,0,0,4,3,1,2,2,1,1,1,0,0,10,4,5,3,4,1,3,1,1,0,0,4,3,3,3, %U A143620 2,2,1,1,1,1,0,0 %N A143620 Triangle read by rows, square of A054521, 1<=k<=n. %C A143620 Left border = phi(n), A000010. %C A143620 Row sums = A053570: (1, 1, 2, 3, 6, 5, 12, 13,...). %F A143620 Triangle read by rows, square of A054521, 1<=k<=n %e A143620 First few rows of the triangle = %e A143620 1; %e A143620 1, 0; %e A143620 2, 0, 0; %e A143620 2, 1, 0, 0; %e A143620 4, 1, 1, 0, 0; %e A143620 2, 1, 1, 1, 0, 0; %e A143620 6, 2, 2, 1, 1, 0, 0; %e A143620 4, 3, 2, 2, 1, 1, 0, 0; %e A143620 6, 2, 4, 2, 2, 1, 1, 0, 0; %e A143620 .. %Y A143620 Cf. A054521, A000010, A053570. %K A143620 nonn,new %O A143620 1,4 %A A143620 Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 27 2008 %I A143615 %S A143615 1,1,3,3,8,3,14,7,14,8,27,7,35,12 %N A143615 Triangle A054521 * A000005 as a vector. %F A143615 Triangle A054521 * A000005 as a vector; where 1's indicate the relative primes of n by rows, and A000005 = d(n): (1, 2, 2, 3, 2, 4, 2, 4, 3,...) %e A143615 a(8) = 7 since the relative primes of 8 are (1, 3, 5, 7). (d(1) + d(3) + d(5) + d(7)) = 1 + 2 + 2 + 2). Or, a(8) = 7 = (1, 0, 1, 0, 1, 0, 1, 0) dot (1, 2, 2, 3, 2, 4, 2, 4), where (1, 0, 1, 0, 1, 0, 1, 0) = row 8 of triangle A054521 and d(n) = (1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2,...). %e A143615 a(7) = 14 = (1, 1, 1, 1, 1, 1, 0) dot (1, 2, 2, 3, 2, 4, 2) = (d(1) + d(2) + d(3) + d(4) + d(5) + d(6)). %Y A143615 Cf. A054521, A000005, A143614. %K A143615 nonn,new %O A143615 1,3 %A A143615 Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 27 2008 %I A143614 %S A143614 1,1,0,2,1,0,2,0,1,0,4,2,1,1,0,2,0,0,0,1,0,6,3,2,1,1,1,0,4,0,1,0,1,0,1, %T A143614 0,6,3,0,2,1,0,1,1,0,4,0,2,0,0,0,1,0,1,0,10,5,3,2,2,1,1,1,1,1,0,4,0,0,0, %U A143614 1,0,1,0,0,0,1,0,12,6,4,3,2,2,1,1,1,1,1,1,0,6,0,2,0,10,0,0,1,0,10,1,0 %N A143614 Triangle read by rows, A054521 * A051731, 1<=k<=n. %C A143614 Left border = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6,...). %C A143614 Row sums = A143615: (1, 1, 3, 3, 8, 3, 14, 7,...). %F A143614 Triangle read by rows, A054521 * A051731, 1<=k<=n. A054521 records the relative primes of n, indicated by a 1's in row n, 0 otherwise. A051731 = the inverse Mobius transform, in which 1's by rows indicate the divisors of n, 0 otherwise. %e A143614 First few rows of the triangle = %e A143614 1; %e A143614 1, 0; %e A143614 2, 1, 0; %e A143614 2, 0, 1, 0; %e A143614 4, 2, 1, 1, 0; %e A143614 2, 0, 0, 0, 1, 0; %e A143614 6, 3, 2, 1, 1, 1, 0; %e A143614 4, 0, 1, 0, 1, 0, 1, 0; %e A143614 6, 3, 0, 2, 1, 0, 1, 1, 0; %e A143614 .. %Y A143614 Cf. A051731, A054521, A000010, A143615. %K A143614 nonn,tabl,new %O A143614 1,4 %A A143614 Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 27 2008 %I A143613 %S A143613 1,2,0,2,2,0,3,0,3,0,2,2,3,4,0,4,2,0,0,5,0,2,2,3,4,5,6,0,4,0,6,0,5,0,7, %T A143613 0,3,4,0,4,5,0,7,8,0,4,2,6,4,0,0,7,0,9,0,2,2,3,4,5,6,7,8,9,10,0,6,2,3,0, %U A143613 10,0,7,0,0,0,11,2,2,3,4,5,6,7,8,9,10,11,12 %N A143613 Triangle read by rows, A051731 * A127368, 1<=k<=n. %C A143613 Left border = d(n). %C A143613 Row sums = A057661: (1, 2, 4, 6, 11, 11, 22, 22, 31,...). %F A143613 Triangle read by rows, A051731 * A127368, 1<=k<=n. A051731 = the inverse Mobius transform. A127368 records the reduced residue system mod n, by rows. %e A143613 First few rows of the triangle = %e A143613 1; %e A143613 2, 0; %e A143613 2, 2, 0; %e A143613 3, 0, 3, 0; %e A143613 2, 2, 3, 4, 0; %e A143613 4, 2, 0, 0, 5, 0; %e A143613 2, 2, 3, 4, 5, 6, 0; %e A143613 .. %Y A143613 Cf. A127368, A051731, A057661, A000005. %K A143613 nonn,tabl,new %O A143613 1,2 %A A143613 Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 27 2008 %I A143612 %S A143612 1,1,0,3,2,0,4,3,3,0,10,9,7,4,0,6,5,5,5,5,0,21,20,18,15,11,6,0,16,15,15, %T A143612 12,12,7,7,0,27,26,24,24,20,15,15,8,0,20,19,19,16,16,16,16,9,9,0,55,54, %U A143612 52,49,45,40,34,27,19,10,0,24,23,23,23,23,18,18,11,11,11,11,0,78,77,75 %N A143612 Triangle read by rows, A127368 * A000012, 1<=k<=n. %C A143612 Left border of the triangle = A123896: (1, 1, 3, 4, 10, 6, 21,...). %C A143612 Row sums = A053818: (1, 1, 5, 10, 30, 26, 91,...). %F A143612 Triangle read by rows, A127368 * A000012, 1<=k<=n. Triangle A127368 records the reduced residue system mod n. The operator A000012 takes partial sums starting from the right in A127368 rows. %e A143612 First few rows of the triangle = %e A143612 1; %e A143612 1, 0; %e A143612 3, 2, 0; %e A143612 4, 3, 3, 0; %e A143612 10, 9, 7, 4, 0; %e A143612 6, 5, 5, 5, 5, 0; %e A143612 21, 20, 18, 15, 11, 6, 0; %e A143612 16, 15, 15, 12, 12, 7, 7, 0; %e A143612 .. %e A143612 Row 5 = (10, 9, 7, 4, 0) since row 5 of triangle A127368 = (1, 2, 3, 4, 0). %Y A143612 Cf. A127368, A023896, A053818. %K A143612 nonn,tabl,new %O A143612 1,4 %A A143612 Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 27 2008 %I A143598 %S A143598 1,2,28,1176,103440,15726880,3684098496,1232799974784,558670427013376, %T A143598 329559835063067136,245462725323910487040,225319148634038399801344, %U A143598 249936012383478860884217856,329609037187846742271984869376 %N A143598 E.g.f.: A(x) = exp(x*sinh(x*G(x))) where G(x) = cosh(x*G(x)) is the e.g.f. of A143601. %F A143598 E.g.f.: A(x) = exp(x*F(x)) where F(x) is the e.g.f. of A007106. %F A143598 E.g.f.: A(x) = sqrt(H(x)*H(-x)) where H(x) = exp(x*sqrt(H(x)/H(-x))) is the e.g.f. of A143599. %e A143598 E.g.f.: A(x) = 1 + 2*x^2/2! + 28*x^4/4! + 1176*x^6/6! + 103440*x^8/8! +... %e A143598 A(x) = exp(x*F(x)) where F(x) = e.g.f. of A007106: %e A143598 F(x) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + 686080*x^9/9! +... %e A143598 A(x) = exp(x*sqrt(G(x)^2 - 1)) where G(x) = e.g.f. of A143601: %e A143598 G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +... %e A143598 A(x) = sqrt(H(x)*H(-x)) where H(x) = e.g.f. of A143599: %e A143598 H(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 53*x^4/4! + 316*x^5/5! +... %o A143598 (PARI) {a(n)=local(G=1+x*O(x^n));for(i=0,n,G=cosh(x*G));n!*polcoeff(exp(x*sqrt(G^2-1)),n)} %Y A143598 Cf. A058014, A143600, A143601, A007106. %K A143598 nonn,new %O A143598 0,2 %A A143598 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 27 2008 %I A143599 %S A143599 1,1,3,10,53,316,2527,22072,239689,2774800,38284091,553477024, %T A143599 9284250109,161180444608,3187413648343,64638167906176,1473221217774353, %U A143599 34190645940363520,882759869810501491,23079229227696318976 %N A143599 E.g.f. satisfies: A(x) = exp( x*sqrt(A(x)/A(-x)) ). %F A143599 E.g.f.: A(x) = exp(x*exp(x*G(x))) where G(x) = cosh(x*G(x)) = e.g.f. of A143601. %F A143599 E.g.f.: sqrt(A(x)/A(-x)) = F(x) = exp(x*[F(x) + 1/F(x)]/2) = e.g.f. of A058014. %F A143599 E.g.f.: [sqrt(A(x)/A(-x)) + sqrt(A(-x)/A(x))]/2 = e.g.f. of A143601. %F A143599 E.g.f.: [sqrt(A(x)/A(-x)) - sqrt(A(-x)/A(x))]/2 = e.g.f. of A007106. %F A143599 E.g.f.: A(x) = H(x/2)^2 where H(x) = exp(x*H(x)/H(-x)) = e.g.f. of A143600. %e A143599 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 53*x^4/4! + 316*x^5/5! +... %e A143599 F(x) = sqrt(A(x)/A(-x)) = e.g.f. of A058014: %e A143599 F(x) = 1 + x + 1*x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! +... %e A143599 where F(x) = exp(x*(F(x) + 1/F(x))/2). %e A143599 G(x) = [sqrt(A(x)/A(-x)) + sqrt(A(-x)/A(x))]/2 = e.g.f. of A143601: %e A143599 G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +... %e A143599 where G(x) = cosh(x*G(x)). %e A143599 S(x) = [sqrt(A(x)/A(-x)) - sqrt(A(-x)/A(x))]/2 = e.g.f. of A007106: %e A143599 S(x) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + 686080*x^9/9! +... %e A143599 where S(x) = sqrt(G(x)^2 - 1) and G(x) = e.g.f. of A143601. %o A143599 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=exp(x*sqrt(A/subst(A,x,-x))));n!*polcoeff(A,n)} %Y A143599 Cf. A058014, A143600, A143601, A007106. %K A143599 nonn,new %O A143599 0,3 %A A143599 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 27 2008 %I A143610 %S A143610 72,108,200,392,500,675,968,1125,1323,1352,1372,2312,2888,3087,3267, %T A143610 4232,4563,5324,6125,6728,7688,7803,8575,8788,9747,10952,11979,13448, %U A143610 14283,14792,15125,17672,19652,19773,21125,22472,22707,25947,27436 %N A143610 Numbers of the form p^2*q^3, where p,q are distinct primes. %C A143610 Also: numbers with prime signature {3,2}. %C A143610 This is a subsequence of A114128. %H A143610 Project Euler, Problem 200. %e A143610 The first elements of this sequence are 3^2*2^3=72, 2^2*3^3=108, 5^2*2^3=200. %o A143610 (PARI) for(n=1, 10^5, omega(n)==2|next; vecsort(factor(n)[,2])==[2,3]~ & print1(n",")) %Y A143610 Cf. A114128. %K A143610 easy,nonn,new %O A143610 1,1 %A A143610 M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Aug 27 2008 %I A143602 %S A143602 1,1,1,7,11,741,14129,521263,20968359,1063764649,63316356389, %T A143602 4408796480331,352958649497387,32158017135672013,3302679619545572265, %U A143602 379346145007147112551,48397471256028983134799 %V A143602 1,1,1,7,-11,741,-14129,521263,-20968359,1063764649,-63316356389,4408796480331, %W A143602 -352958649497387,32158017135672013,-3302679619545572265,379346145007147112551, %X A143602 -48397471256028983134799,6817654800019973404119633,-1054828080584161260522077645 %N A143602 E.g.f. satisfies: A(x) = exp( A(x)*Series_Reversion[x*A(x)] ). %F A143602 E.g.f. satisfies: A(x*A(x)) = exp(x*A(x*A(x))) = LambertW(-x)/(-x). %e A143602 A(x) = 1 + x + x^2/2! + 7*x^3/3! - 11*x^4/4! + 741*x^5/5! - 14129*x^6/6! +-... %e A143602 A(x*A(x)) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! +... %e A143602 LambertW(-x)/(-x) = 1 + x + 3^1*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! +... %e A143602 log(A(x)) = x + 2*x^3/2! - 9*x^4/3! + 172*x^5/4! - 3205*x^6/5! +-... %e A143602 Series_Reversion[x*A(x)] = x - x^2 + 3*x^3/2! - 22*x^4/3! + 281*x^5/4! - 5396*x^6/5! +-... %o A143602 (PARI) {a(n)=local(A=1);for(i=0,n,A=exp(A*serreverse(x*A+x^2*O(x^n))));n!*polcoeff(A,n)} %Y A143602 Cf. A000272. %K A143602 sign,new %O A143602 0,4 %A A143602 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 26 2008 %I A143601 %S A143601 1,1,13,541,47545,7231801,1695106117,567547087381,257320926233329, %T A143601 151856004814953841,113144789723082206461,103890621918675777804301, %U A143601 115270544419577901796226473,152049571406030636219959644841 %N A143601 Number of labeled odd degree trees with 2n+1 nodes. %F A143601 E.g.f. satisfies: A(x) = cosh(x*A(x)). %F A143601 E.g.f.: A(x) = (1/x)*Series_Reversion( x/cosh(x) ). %F A143601 E.g.f.: sqrt(A(x)^2 - 1) = e.g.f. of A007106. %F A143601 E.g.f.: exp(x*A(x)) = A(x) + sqrt(A(x)^2-1) = e.g.f. of A058014. %F A143601 E.g.f.: A(x) = [F(x) + F(-x)]/2 where F(x) = exp(x*[F(x) + 1/F(x)]/2) = e.g.f. of A058014. %F A143601 E.g.f.: A(2x) = [G(x)/G(-x) + G(-x)/G(x)]/2 where G(x) = exp(x*G(x)/G(-x)) = e.g.f. of A143600. %e A143601 E.g.f.: A(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +... %e A143601 The e.g.f. of A007106 (a bisection of A058014) is given by: %e A143601 sqrt(A(x)^2 - 1) = x + 4*x^3/3! + 96*x^5 + 5888*x^7 + 686080*x^9/9! +... %e A143601 The e.g.f. of A058014 is given by: %e A143601 F(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! +... %e A143601 where A(x) = [F(x) + F(-x)]/2 and exp(x*A(x)) = F(x). %e A143601 The e.g.f. of A143600 is given by: %e A143601 G(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 249*x^4/4! + 2561*x^5/5! +... %e A143601 where A(2x) = [G(x)/G(-x) + G(-x)/G(x)]/2. %o A143601 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=cosh(x*A));n!*polcoeff(A,n)} %Y A143601 Cf. A058014, A143600, A007106. %K A143601 nonn,new %O A143601 0,3 %A A143601 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 26 2008 %I A143600 %S A143600 1,1,5,25,249,2561,40573,641817,14110001,302279617,8530496181, %T A143600 230851019609,7964867290537,260618470319169,10635790073585069, %U A143600 408342804482252761,19246730825243728737,848289638051491455617 %N A143600 E.g.f. satisfies: A(x) = exp(x*A(x)/A(-x)). %F A143600 E.g.f.: A(x) = exp(x*exp(2x*G(2x))) where G(x) = cosh(x*G(x)) = e.g.f. of A143601. %F A143600 E.g.f.: [A(x)/A(-x) + A(-x)/A(x)]/2 = G(2x) where G(x) = cosh(x*G(x)) = e.g.f. of A143601. %F A143600 E.g.f.: A(x)/A(-x) = exp(x*[A(x)/A(-x) + A(-x)/A(x)]) = F(2x) where F(x) = exp(x*[F(x) + 1/F(x)]/2) = e.g.f. of A058014. %e A143600 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 249*x^4/4! + 2561*x^5/5! +... %e A143600 A(x)/A(-x) = F(2x) where F(x) is the e.g.f. of A058014: %e A143600 A(x)/A(-x) = 1 + 2*x + 4*x^2/2! + 32*x^3/3! + 208*x^4/4! + 3072*x^5/5! +... %e A143600 F(x) = 1 + x + 1*x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! +... %e A143600 which satisfies: F(x) = exp(x*(F(x) + 1/F(x))/2). %e A143600 (A(x)/A(-x) + A(-x)/A(x))/2 = G(2x) where G(x) is the e.g.f. of A143601: %e A143600 (A(x)/A(-x) + A(-x)/A(x))/2 = 1 + 4*x^2/2! + 208*x^4/4! + 34624*x^6/6! +... %e A143600 G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +... %e A143600 which satisfies G(x) = cosh(x*G(x)). %o A143600 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=exp(x*A/subst(A,x,-x)));n!*polcoeff(A,n)} %Y A143600 Cf. A058014, A143601, A007106. %K A143600 nonn,new %O A143600 0,3 %A A143600 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 26 2008 %I A143956 %S A143956 1,3,2,5,4,2,7,6,4,2,9,8,6,4,2,11,10,8,6,4,2,13,12,10,8,6,4,2,15,14,12, %T A143956 10,8,6,4,2,17,16,14,12,10,8,6,4,2,19,18,16,14,12,10,8,6,4,2,21,20,18, %U A143956 16,14,12,10,8,6,4,2 %N A143956 Triangle read by rows, A000012 * A136521 * A000012; 1<=k<=n. %C A143956 Row sums = A028387: (1, 5, 11, 19, 29, 41,...) %F A143956 Triangle read by rows, A000012 * A136521 * A000012; 1<=k<=n. A136521 = an infinite lower triangular matrix with (1,2,2,2,...) in the main diagonal the the rest zeros. %e A143956 First few rows of the triangle = %e A143956 1; %e A143956 3, 2; %e A143956 5, 4, 2; %e A143956 7, 6, 4, 2; %e A143956 9, 8, 6, 4, 2; %e A143956 11, 10, 8, 6, 4, 2; %e A143956 13, 12, 10, 8, 6, 4, 2; %e A143956 15, 14, 12, 10, 8, 6, 4, 2; %e A143956 .. %Y A143956 Cf. A136521, A028387. %K A143956 nonn,tabl,new %O A143956 1,2 %A A143956 Gary W. Adamson & Roger L. Bagula (qntmpkt(AT)yahoo.com), Aug 26 2008 %I A143595 %S A143595 1,2,2,3,4,2,4,6,4,2,5,8,6,4,2,6,10,8,6,4,2,7,12,10,8,6,4,2,8,14,12,10, %T A143595 8,6,4,2,9,16,14,12,10,8,6,4,2,10,18,16,14,12,10,8,6,4,2,11,20,18,16,14, %U A143595 12,10,8,6,4,2 %N A143595 Triangle read by rows, A000012 * (an infinite lower triangular matrix with 1's in the first column and the rest 2's); 1<=k<=n. %C A143595 Row sums = n^2. %C A143595 The linear sequence A056944 is more appropriately a triangle, (reversal of A143595). %F A143595 Triangle read by rows, A000012 * (an infinite lower triangular matrix with 1's in the first column and the rest 2's); i.e. (1; 1,2; 1,2,2;...). A000012 = an infinite lower triangular matrix with all 1's. 1<=k<=n %e A143595 First few rows of the triangle = %e A143595 1; %e A143595 2, 2; %e A143595 3, 4, 2; %e A143595 4, 6, 4, 2; %e A143595 5, 8, 6, 4, 2; %e A143595 6, 10, 8, 6, 4, 2; %e A143595 7, 12, 10, 8, 6, 4, 2; %e A143595 .. %Y A143595 Cf. A056944. %K A143595 nonn,tabl,new %O A143595 1,2 %A A143595 Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 26 2008 %I A143594 %S A143594 1,2,2,2,2,2,3,4,2,2,2,2,2,2,2,4,6,4,2,2,2,2,2,2,2,2,2,2,4,6,4,4,2,2,2, %T A143594 2,3,4,4,2,2,2,2,2,2,4,6,4,4,4,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,6,10,8,6, %U A143594 4,4,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 %N A143594 Triangle read by rows, A051731 * (an infinite lower triangular matrix with 1's in the first column and the rest 2's). %C A143594 Left column = d(n), A000005: (1, 2, 2, 3, 2, 4, 2,...). %C A143594 Prime rows have all 2's. %C A143594 Row sums = A129235: (1, 4, 6, 11, 10, 10,...). %F A143594 Triangle read by rows, A051731 * (an infinite lower triangular matrix with 1's in the first column and the rest 2's), i.e. (1; 1,2; 1,2,2;...), 1<=k<=n A051731 = the inverse Mobius transform. %e A143594 First few rows of the triangle = %e A143594 1; %e A143594 2, 2; %e A143594 2, 2, 2; %e A143594 3, 4, 2, 2; %e A143594 2, 2, 2, 2, 2; %e A143594 4, 6, 4, 2, 2, 2; %e A143594 2, 2, 2, 2, 2, 2, 2; %e A143594 .. %Y A143594 Cf. A129235, A000005, A051731. %K A143594 nonn,tabl,new %O A143594 1,2 %A A143594 Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 26 2008 %I A143593 %S A143593 1,3,4,5,8,4,7,12,8,4,9,16,12,8,4,11,20,16,12,8,4,13,24,20,16,12,8,4,15, %T A143593 28,24,20,16,12,8,4,17,32,28,24,20,16,12,8,4,19,36,32,28,24,20,16,12,8, %U A143593 4 %N A143593 Triangle read by rows, square of an infinite lower triangular matrix with 1's in the first column and the rest 2's. %C A143593 Row sums = A056220 starting with offset 1: (1, 7, 17, 31, 49, 71, 97,...). %F A143593 Triangle read by rows, square of an infinite lower triangular matrix with 1's in the first column and the rest 2's. Square of (A000012 * (S(k) * 0^(n-k)), 1<=k<=n %e A143593 The square of the infinite lower triangular matrix (1; 1,2; 1,2,2;...) = %e A143593 1; %e A143593 3, 4; %e A143593 5, 8, 4; %e A143593 7, 12, 8, 4; %e A143593 9, 16, 12, 8, 4; %e A143593 11, 20, 16, 12, 8, 4; %e A143593 13, 24, 20, 16, 12, 8, 4; %e A143593 .. %Y A143593 Cf. A056220. %K A143593 nonn,tabl,new %O A143593 1,2 %A A143593 Gary W. Adamson & Roger L. Bagula (qntmpkt(AT)yahoo.com), Aug 26 2008 %I A143592 %S A143592 0,1,0,1,1,3,3,6,8,12,18,26,40,60,89,135,201,303,453,680,1022,1528,2299, %T A143592 3443,5176,7750,11642,17443,26172,39246,58874,88308,132461,198694, %U A143592 298094,447097,670645,1005992,1508898,2263540 %N A143592 Number of 2's in row n of A143589 (a Kolakoski fan). %C A143592 A143590 = A143591 + A143592. %e A143592 The first 6 rows of A143589 are 1; 2; 1,1; 2,1; 1,1,2; 2,1,2,2. The %e A143592 respective numbers of 1's are 0,1,0,1,1,3. %Y A143592 Cf. A000002, A143587, A143589, A143590, A143591. %K A143592 nonn,new %O A143592 1,6 %A A143592 Clark Kimberling (ck6(AT)evansville.edu), Aug 25 2008 %I A143591 %S A143591 1,0,2,1,2,1,4,4,8,12,18,28,40,60,91,134,203,302,455,681,1019,1535,2292, %T A143591 3447,5157,7759,11617,17458,26172,39270,58888,88328,132483,198711, %U A143591 298005,447096,670645,1005943,1509029,2263285 %N A143591 Number of 1's in row n of A143589 (a Kolakoski fan). %C A143591 A143590 = A143591 + A143592. %e A143591 The first 6 rows of A143589 are 1; 2; 1,1; 2,1; 1,1,2; 2,1,2,2. The %e A143591 respective numbers of 1's are 1,0,2,1,2,1. %Y A143591 Cf. A000002, A143587, A143589, A143590, A143592. %K A143591 nonn,new %O A143591 1,3 %A A143591 Clark Kimberling (ck6(AT)evansville.edu), Aug 25 2008 %I A143590 %S A143590 1,1,2,2,3,4,7,10,16,24,36,54,80,120,180,269,404,605,908,1361,2041,3063, %T A143590 4591,6890,10333,15509,23259,34901,52344,78516,117762,176636,264944, %U A143590 397405,596099,894193,1341290,2011935,3017927,4526825 %N A143590 Length of row n of A143589 (a Kolakoski fan). %C A143590 Conjecture (following Benoit Cloitre's conjecture at A111090): %C A143590 if L=A143489, then L(n)*(2/3)^n approaches a constant. %e A143590 The first 6 rows of A143589 are 1; 2; 1,1; 2,1; 1,1,2; 2,1,2,2. Their %e A143590 lengths are 1,1,2,2,3,4. %Y A143590 Cf. A000002, A111090, A143477, A143586, A143489. %K A143590 nonn,new %O A143590 1,3 %A A143590 Clark Kimberling (ck6(AT)evansville.edu), Aug 25 2008 %I A143589 %S A143589 1,2,1,1,2,1,1,1,2,2,1,2,2,1,1,2,1,1,2,2,2,1,2,2,1,2,1,1,2,2,1,1,2,1,1, %T A143589 2,2,1,2,2,1,2,1,1,2,2,2,1,2,2,1,2,1,1,2,2,1,2,2,1,1,2,1,1,2,1,2,2,1,1, %U A143589 1,1,2,1,1,2,2,1,2,2,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,1,1,2,1,1,2,2,1 %N A143589 Kolakoski fan based on A000034 with initial row 1. %C A143589 Conjecture (following Benoit Cloitre's conjecture at A111090): if L(n) %C A143589 is the number (assumed finite) of terms in row n of K, then %C A143589 L(n)*(2/3)^n approaches a constant. (L= A143590.) %F A143589 Introduced here is an array K called the "Kolakoski fan based on a sequence s with initial row w": suppose that s=(s(1),s(2),...) is a sequence of 1's and 2's and that w=(w(1),w(2),...) is a finite or infinite sequence of 1's and 2's. Assume that s(1)=w(1) and that if w(1)=1 then s contains at least one 2. Row 1 of the array K is w. Subsequent rows are defined inductively: the first term of row n is s(n), and the remaining terms are defined by Kolakoski substitution; viz., each number in row n-1 tells the string-length (1 or 2) of the next string in row n, each term being either 1 or 2. %e A143589 s=(1,2,1,2,1,2,1,2,...) and w=1, so the first 7 rows are %e A143589 1 %e A143589 2 %e A143589 1 1 %e A143589 2 1 %e A143589 1 1 2 %e A143589 2 1 2 2 %e A143589 1 1 2 1 1 2 2 %Y A143589 Cf. A000002, A143477, A143490. %K A143589 nonn,tabf,new %O A143589 1,2 %A A143589 Clark Kimberling (ck6(AT)evansville.edu), Aug 25 2008 %I A143588 %S A143588 0,1,2,2,3,4,6,10,15,22,33,52,74,113,168,255,377,567,850,1280,1916,2878, %T A143588 4324,6473,9720,14577,21862,32766,49150,73758,110640,165962,249013, %U A143588 373405,560201,840358,1260296,1890539,2835987,4253676,6380430,9570746 %N A143588 Number of 2's in row n of the Kolakoski fan A143477. %C A143588 A143586 = A143587 + A143588. %e A143588 The first 4 rows of A143477 are 1; 2; 22; 1122. The respective numbers of 1's are 0,1,2,2. %Y A143588 Cf. A000002, A143477, A143486, A143487. %K A143588 nonn,new %O A143588 1,3 %A A143588 Clark Kimberling (ck6(AT)evansville.edu), Aug 25 2008 %I A143587 %S A143587 1,0,0,2,3,5,7,9,14,22,33,47,77,112,170,251,384,571,855,1275,1919,2873, %T A143587 4305,6480,9706,14569,21861,32819,49201,73743,110619,165937,248848, %U A143587 373469,560078,840122,1260542,1890595,2835686,4253984,6380906,9571020 %N A143587 Number of 1's in row n of the Kolakoski fan A143477. %C A143587 A143586 = A143587 + A143588. %e A143587 The first 4 rows of A143477 are 1; 2; 22; 1122. The respective numbers of 1's are 1,0,0,2. %Y A143587 Cf. A000002, A143477, A143486, A143588. %K A143587 nonn,new %O A143587 1,4 %A A143587 Clark Kimberling (ck6(AT)evansville.edu), Aug 25 2008 %I A143586 %S A143586 1,1,2,4,6,9,13,19,29,44,66,99,151,225,338,506,761,1138,1705,2555,3835, %T A143586 5751,8629,12953,19426,29146,43723,65585,98351,147501,221259,331899, %U A143586 497861,746874,1120279,1680480,2520838,3781134,5671673,8507660,12761336 %N A143586 Length of row n of the Kolakoski fan A143477. %C A143586 Conjecture: if L(n)=A143586(n), then lim(L(n+1)/L(n)) = 3/2. For a more general conjecture see A143589. %e A143586 The first 4 rows of A143477 are %e A143586 1 %e A143586 2 %e A143586 22 %e A143586 1122. Their lengths are 1,1,2,4. %Y A143586 Cf. A000002, A143477. %K A143586 nonn,new %O A143586 1,3 %A A143586 Clark Kimberling (ck6(AT)evansville.edu), Aug 25 2008 %I A143584 %S A143584 11,23,25,28,29,35,36,37,39 %N A143584 a(n) is the multiplicative order of 2 modulo some overpseudoprime to base 2. %C A143584 A064078(a(n)) is composite number. The sequence have a positive density since %C A143584 contains, in particular, numbers of the form 8n+20 for n>=1 (C.Pomerance, private correspondence). Since, e. g., 38 is not in the sequence, then there is not an overpseudoprime m such that ord_m(2)=38. %Y A143584 Cf. A141232 A002326 A064078 A122929. %K A143584 nonn,new %O A143584 1,1 %A A143584 Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 25 2008 %I A143583 %S A143583 1,12,164,2352,34596,516912,7806224,118803648,1818757924,27972399792, %T A143583 431824158864,6686855325888,103814819552016,1615296581684928, %U A143583 25180747436810304,393189646497706752,6148451986328464164 %N A143583 Apery-like numbers: a(n) = 1/C(2n,n)*sum {k = 0..n} C(2k,k)*C(4k,2k)*C(2n-2k,n-k)*C(4n-4k,2n-2k). %C A143583 These numbers bear some analogy to the Apery numbers A005258. They appear in the evaluation of the spectral zeta function of the non-commutative harmonic oscillator zeta_Q(s) at s = 2, and satisfy a recurrence relation similar to the one satisfied by the Apery numbers. %H A143583 K. Kimoto and M. Wakayama, Apery-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators Kyushu J. Math. Vol. 60, 2006, 383-404. %F A143583 a(n) = 1/C(2n,n)*sum {k = 0..n} C(2k,k)*C(4k,2k)*C(2n-2k,n-k)*C(4n-4k,2n-2k). Recurrence relation: a(0) = 1, a(1) = 12, n^2*a(n) = 4*(8*n^2-8*n+3)*a(n-1) – 256*(n-1)^2*a(n-2). Congruences: For odd prime p, a(m*p^r) = a(m*p^(r-1)) (mod p^r) for any m,r in N. %p A143583 a := n -> 1/binomial(2n,n)*add(binomial(2k,k)*binomial(4k,2k)*binomial(2n-2k,n-k)*binomial(4n-4k,2n-2k),k = 0..n): seq(a(n),n = 0..20): %Y A143583 Cf. A005258. %K A143583 easy,nonn,new %O A143583 0,2 %A A143583 Peter Bala (pbala(AT)toucansurf.com), Aug 25 2008 %I A143499 %S A143499 1,8,1,72,18,1,720,270,30,1,7920,3960,660,44,1,95040,59400,13200,1320, %T A143499 60,1,1235520,926640,257400,34320,2340,78,1,17297280,15135120,5045040, %U A143499 840840,76440,3822,98,1,259459200,259459200,100900800,20180160,2293200 %N A143499 Triangle of unsigned 4-restricted Lah numbers. %C A143499 This is the case r = 4 of the unsigned r-restricted Lah numbers L(r;n,k). The unsigned 4-restricted Lah numbers count the number of partitions of the set {1,2,...,n} into k ordered lists with the restriction that the elements 1, 2, 3 and 4 belong to different lists. For other cases see A105278 (r = 1), A143497 (r = 2 and comments on the general case) and A143498 (r = 3). %C A143499 The unsigned 4-restricted Lah numbers are related to the 4-restricted Stirling numbers: the lower triangular array of unsigned 4-restricted Lah numbers may be expressed as the matrix product St1(4) * St2(4), where St1(4) = A143493 and St2(4) = A143496 are the arrays of 4-restricted Stirling numbers of the first and second kind respectively. An alternative factorization for the array is as St1 * P^6 * St2, where P denotes Pascal’s triangle, A007318, St1 is the triangle of unsigned Stirling numbers of the first kind, abs(A008275), and St2 denotes the triangle of Stirling numbers of the second kind, A008277. %H A143499 Neuwirth Erich, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 No. 1-3, 33-51 (2001) %F A143499 T(n,k) = (n-4)!/(k-4)!*binomial(n+3,k+3), n,k >= 4. Recurrence: T(n,k) = (n+k-1)*T(n-1,k) + T(n-1,k-1) for n,k >= 4, with the boundary conditions: T(n,k) = 0 if n < 4 or k < 4; T(4,4) = 1. E.g.f. for column k: sum {n >= k} T(n,k)*t^n/(n-4)! = 1/(k-4)!*t^k/(1-t)^(k+4) for k >= 4. E.g.f: sum {n = 4..inf} sum {k = 4..n} T(n,k)*x^k*t^n/(n-4)! = (x*t)^4/(1-t)^8*exp(x*t/(1-t)) = (x*t)^4*(1 + (8+x)t +(72+18x+x^2)t^2/2! + ... ). Generalized Lah identity: (x+7)*(x+8)*...*(x+n+2) = sum {k = 4..n} T(n,k)*(x-1)*(x-2)*...*(x-k+4). The polynomials 1/n!*sum {k = 4..n+4} T(n+4,k)*(-x)^(k-4) for n >= 0 are generalized Laguerre polynomials Laguerre(n,7,x). Array = A132493* A143496 = abs(A008275) * ( A007318 )^6 * A008277 (apply Theorem 10 of [Neuwirth]). Array equals exp(D), where D is the array with the quadratic sequence (8,18,30,44, ... ) on the main subdiagonal and zeros everywhere else. %e A143499 Triangle begins %e A143499 n\k|......4......5......6......7......8......9 %e A143499 ============================================== %e A143499 4..|......1 %e A143499 5..|......8......1 %e A143499 6..|.....72.....18......1 %e A143499 7..|....720....270.....30......1 %e A143499 8..|...7920...3960....660.....44......1 %e A143499 9..|..95040..59400..13200...1320.....60......1 %e A143499 .. %e A143499 T(5,4) = 8. The partitions of {1,2,3,4,5} into 4 ordered lists, such that the elements 1, 2, 3 and 4 lie in different lists, are: {1}{2}{3}{4,5} and {1}{2}{3}{5,4}, {1}{2}{4}{3,5} and {1}{2}{4}{5,3}, {1}{3}{4}{2,5} and {1}{3}{4}{5,2}, {2}{3}{4}{1,5} and {2}{3}{4}{5,1}. %p A143499 with combinat: T := (n, k) -> (n-4)!/(k-4)!*binomial(n+3,k+3): for n from 4 to 13 do seq(T(n, k), k = 4..n) end do; %Y A143499 Cf. A007318, A008275, A008277, A049388 (column 4), A105278 (unsigned Lah numbers), A143493, A143496, A143497, A143498. %K A143499 easy,nonn,tabl,new %O A143499 4,2 %A A143499 Peter Bala (pbala(AT)toucansurf.com), Aug 25 2008 %I A143498 %S A143498 1,6,1,42,14,1,336,168,24,1,3024,2016,432,36,1,30240,25200,7200,900,50, %T A143498 1,332640,332640,118800,19800,1650,66,1,3991680,4656960,1995840,415800, %U A143498 46200,2772,84,1,51891840,69189120,34594560,8648640,1201200,96096,4368 %N A143498 Triangle of unsigned 3-restricted Lah numbers. %C A143498 For a signed version of this triangle see A062138. This is the case r = 3 of the unsigned r-restricted Lah numbers L(r;n,k). The unsigned 3-restricted Lah numbers count the number of partitions of the set {1,2,...,n} into k ordered lists with the restriction that the elements 1, 2 and 3 belong to different lists. For other cases see A105278 (r = 1), A143497 (r = 2, and comments on the general case) and A143499 (r = 4). %C A143498 The unsigned 3-restricted Lah numbers are related to the 3-restricted Stirling numbers: the lower triangular array of unsigned 3-restricted Lah numbers may be expressed as the matrix product St1(3) * St2(3), where St1(3) = A143492 and St2(3) = A143495 are the arrays of 3-restricted Stirling numbers of the first and second kind respectively. An alternative factorization for the array is as St1 * P^4 * St2, where P denotes Pascal’s triangle, A007318, St1 is the triangle of unsigned Stirling numbers of the first kind, abs(A008275), and St2 denotes the triangle of Stirling numbers of the second kind, A008277. %H A143498 Neuwirth Erich, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 No. 1-3, 33-51 (2001) %F A143498 T(n,k) = (n-3)!/(k-3)!*binomial(n+2,k+2) for n,k >= 3. Recurrence: T(n,k) = (n+k-1)*T(n-1,k) + T(n-1,k-1) for n,k >= 3, with the boundary conditions: T(n,k) = 0 if n < 3 or k < 3; T(3,3) = 1. E.g.f. for column k: sum {n >= k} T(n,k)*t^n/(n-3)! = 1/(k-3)!*t^k/(1-t)^(k+3) for k >= 3. E.g.f: sum {n = 3..inf} sum {k = 3..n} T(n,k)*x^k*t^n/(n-3)! = (x*t)^3/(1-t)^6*exp(x*t/(1-t)) = (x*t)^3*(1 + (6+x)t +(42+14x+x^2)t^2/2! + ... ). Generalized Lah identity: (x+5)*(x+6)*...*(x+n+1) = sum {k = 3..n} T(n,k)*(x-1)*(x-2)*...*(x-k+3). The polynomials 1/n!*sum {k = 3..n+3} T(n+3,k)*(-x)^(k-3) for n >= 0 are generalized Laguerre polynomials Laguerre(n,5,x). See A062138. Array = A143492 * A143495 = abs(A008275) * ( A007318 )^4 * A008277 (apply Theorem 10 of [Neuwirth]). Array equals exp(D), where D is the array with the quadratic sequence (6,14,24,36, ... ) on the main subdiagonal and zeros everywhere else. %e A143498 Triangle begins %e A143498 n\k|......3......4......5......6......7......8 %e A143498 ============================================== %e A143498 3..|......1 %e A143498 4..|......6......1 %e A143498 5..|.....42.....14......1 %e A143498 6..|....336....168.....24......1 %e A143498 7..|...3024...2016....432.....36......1 %e A143498 8..|..30240..25200...7200....900.....50......1 %e A143498 .. %e A143498 T(4,3) = 6. The partitions of {1,2,3,4} into 3 ordered lists, such that the elements 1, 2 and 3 lie in different lists, are: {1}{2}{3,4} and {1}{2}{4,3}, {1}{3}{2,4} and {1}{3}{4,2}, {2}{3}{1,4} and {2}{3}{4,1}. %p A143498 with combinat: T := (n, k) -> (n-3)!/(k-3)!*binomial(n+2,k+2): for n from 3 to 12 do seq(T(n, k), k = 3..n) end do; %Y A143498 Cf. A001725 (column 3), A007318, A008275, A008277, A062138, A062148 – A062152 (column 4 to column 8), A062191 (alt. row sums), A062192 (row sums), A105278 (unsigned Lah numbers), A143492, A143495, A143497, A143499. %K A143498 easy,nonn,tabl,new %O A143498 3,2 %A A143498 Peter Bala (pbala(AT)toucansurf.com), Aug 25 2008 %I A143497 %S A143497 1,4,1,20,10,1,120,90,18,1,840,840,252,28,1,6720,8400,3360,560,40,1, %T A143497 60480,90720,45360,10080,1080,54,1,604800,1058400,635040,176400,25200, %U A143497 1890,70,1,6652800,13305600,9313920,3104640,554400,55440,3080,88,1 %N A143497 Triangle of unsigned 2-restricted Lah numbers. %C A143497 For a signed version of this triangle see A062137. The unsigned 2-restricted Lah number L(2;n,k) counts the number of partitions of the set {1,2,...,n} into k ordered lists with the restriction that the elements 1 and 2 must belong to different lists. More generally, the unsigned r-restricted Lah number L(r;n,k) counts the number of partitions of the set {1,2,...,n} into k ordered lists with the restriction that the elements 1, 2, ..., r belong to different lists. If r = 1 there is no restriction and we obtain the unsigned Lah numbers A105278. For other cases see A143498 (r = 3) and A143499 (r = 4). We make some remarks on the general case. %C A143497 The unsigned restricted Lah numbers occur as connection constants in the generalized Lah identity (x+2r-1)*(x+2r)*...*(x+2r+n-r-2) = sum {k = r..n} L(r;n,k)*(x-1)*(x-2)*...*(x-k+r) for n >=r, and where any empty products are taken equal to 1 (for a bijective proof of the identity, follow the proof of [Petkovsek and Pisanski] but restrict r of the Argonauts to different paths). %C A143497 The unsigned restricted Lah numbers satisfy the same recurrence as the unsigned Lah numbers, namely, L(r;n,k) = (n+k-1)*L(r;n-1,k) + L(r;n-1,k-1), but with the boundary conditions: L(r;n,k) = 0 if n < r or if k < r; L(r;r,r) = 1. The recurrence has the explicit solution L(r;n,k) = (n-r)!/(k-r)!*binomial(n+r-1,k+r-1) for n,k >= r. It follows that the unsigned r-restricted Lah numbers have ‘vertical’ generating functions for k >= r of the form sum {n >= k} L(r;n,k)*t^n/(n-r)! = 1/(k-r)!*t^k/(1-t)^(k+r). This yields the e.g.f. for the array of unsigned r-restricted Lah numbers in the form: sum {n,k >= r} L(r;n,k)*x^k*t^n/(n-r)! = (x*t)^r * 1/(1-t)^(2r) * exp(x*t/(1-t)) = (x*t)^r (1 + (2r+x)*t + (2r*(2r+1) + 2*(2r+1)*x + x^2)*t^2/2! + ... ). The array of unsigned r-restricted Lah numbers begins %C A143497 1...................0..................0.............0... %C A143497 2r..................1..................0.............0... %C A143497 2r*(2r+1)...........2*(2r+1)...........1.............0... %C A143497 2r*(2r+1)*(2r+2)....3*(2r+1)*(2r+2)....3*(2r+2)......1... %C A143497 .. %C A143497 and equals exp(D(r)), where D(r) is the array with the sequence (2*r, 2*(2r+1), 3*(2r+2), 4*(2r+3), ... ) on the main subdiagonal and zeros everywhere else. %C A143497 The unsigned r-restricted Lah numbers are related to the r-restricted Stirling numbers: the lower triangular array of unsigned r-restricted Lah numbers may be expressed as the matrix product St1(r) * St2(r), where St1(r) and St2(r) denote the arrays of r-restricted Stirling numbers of the first and second kind respectively. The theory of restricted Stirling numbers is developed in [Broder]. See A143491 – A143496 for tables of restricted Stirling numbers. An alternative factorization for the array is as St1 * P^(2r-2) * St2, where P denotes Pascal’s triangle, A007318, St1 is the triangle of unsigned Stirling numbers of the first kind, abs(A008275), and St2 denotes the triangle of Stirling numbers of the second kind, A008277 (apply Theorem 10 of [Neuwirth]). %H A143497 Broder Andrei Z., The r-Stirling numbers, Discrete Math. 49, 241-259 (1984) %H A143497 Neuwirth Erich, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 No. 1-3, 33-51 (2001) %H A143497 Marko Petkovsek, Tomaz Pisanski, Combinatorial interpretation of unsigned Stirling and Lah numbers %F A143497 T(n,k) = (n-2)!/(k-2)!*C(n+1,k+1), n,k >= 2. Recurrence: T(n,k) = (n+k-1)*T(n-1,k) + T(n-1,k-1) for n,k >= 2, with the boundary conditions: T(n,k) = 0 if n < 2 or k < 2; T(2,2) = 1. E.g.f. for column k: sum {n >= k} T(n,k)*t^n/(n-2)! = 1/(k-2)!*t^k/(1-t)^(k+2) for k >= 2. E.g.f: sum {n = 2..inf} sum {k = 2..n} T(n,k)*x^k*t^n/(n-2)! = (x*t)^2/(1-t)^4*exp(x*t/(1-t)) = (x*t)^2*(1 + (4+x)t +(20+10x+x^2)t^2/2! + ... ). Generalized Lah identity: (x+3)*(x+4)*...*(x+n) = sum {k = 2..n} T(n,k)*(x-1)*(x-2)*...*(x-k+2). The polynomials 1/n!*sum {k = 2..n+2} T(n+2,k)*(-x)^(k-2) for n >= 0 are generalized Laguerre polynomials Laguerre(n,3,x). See A062137. Array = A143491 * A143494 = abs(A008275) * ( A007318 )^2 * A008277 (apply Theorem 10 of [Neuwirth]). Array equals exp(D), where D is the array with the quadratic sequence (4,10,18,28, ... ) on the main subdiagonal and zeros elsewhere. %e A143497 Triangle begins %e A143497 n\k|.....2.....3.....4.....5.....6.....7 %e A143497 ======================================== %e A143497 2..|.....1 %e A143497 3..|.....4.....1 %e A143497 4..|....20....10.....1 %e A143497 5..|...120....90....18.....1 %e A143497 6..|...840...840...252....28.....1 %e A143497 7..|..6720..8400..3360...560....40.....1 %e A143497 .. %e A143497 T(4,3) = 10. The partitions of {1,2,3,4} into 3 ordered lists such that the elements 1 and 2 lie in different lists are: {1}{2}{3,4} and {1}{2}{4,3}, {1}{3}{2,4} and {1}{3}{4,2}, {1}{4}{2,3} and {1}{4}{3,2}, {2}{3}{1,4} and {2}{3}{4,1}, {2}{4}{1,3} and {2}{4}{3,1}. The remaining two partitions {3}{4}{1,2} and {3}{4}{2,1} are not allowed because the elements 1 and 2 belong to the same block. %p A143497 with combinat: T := (n, k) -> (n-2)!/(k-2)!*binomial(n+1,k+1): for n from 2 to 11 do seq(T(n, k), k = 2..n) end do; %Y A143497 Cf. A001715 (column 2), A007318, A008275, A008277, A061206 (column 3), A062137, A062141 – A062144 ( column 4 to column 7), A062146 (alt. row sums), A062147 (row sums), A105278 (unsigned Lah numbers), A143491, A143494, A143498, A143499. %K A143497 easy,nonn,tabl,new %O A143497 2,2 %A A143497 Peter Bala (pbala(AT)toucansurf.com), Aug 25 2008 %I A143582 %S A143582 1,3,20,56,576,1408,6656,5120,278528,622592,2752512,6029312,52428800, %T A143582 113246208,486539264,1040187392,11811160064,15032385536,317827579904, %U A143582 223338299392,5634997092352,11819749998592,3298534883328 %N A143582 Denominators of coefficient of x^(n+1/2) in the series expansion of the haversine. %H A143582 Eric Weisstein's World of Mathematics, Inverse Haversine %e A143582 2*Sqrt[z] + z^(3/2)/3 + (3*z^(5/2))/20 + (5*z^(7/2))/56 + (35*z^(9/2))/576 + ... %Y A143582 Cf. A143581. %K A143582 nonn,frac,new %O A143582 1,2 %A A143582 E. W. Weisstein (eric(AT)weisstein.com), Aug 24, 2008 %I A143581 %S A143581 2,1,3,5,35,63,231,143,6435,12155,46189,88179,676039,1300075,5014575, %T A143581 9694845,100180065,116680311,2268783825,1472719325,34461632205, %U A143581 67282234305,17534158031,514589420475,8061900920775,5267108601573 %N A143581 Numerators of coefficient of x^(n+1/2) in the series expansion of the haversine. %H A143581 Eric Weisstein's World of Mathematics, Inverse Haversine %e A143581 2*Sqrt[z] + z^(3/2)/3 + (3*z^(5/2))/20 + (5*z^(7/2))/56 + (35*z^(9/2))/576 + ... %Y A143581 Cf. A143582. %K A143581 nonn,frac,new %O A143581 1,1 %A A143581 E. W. Weisstein (eric(AT)weisstein.com), Aug 24, 2008 %I A143580 %S A143580 1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0,1,1, %T A143580 0,1,0,0,1,1,0 %N A143580 A143579 mod 2. %C A143580 Two conjectures: If n is even, the ratio of 1's to 0's = 1:1. %C A143580 There are no three adjacent terms of the same parity. %F A143580 Parity of A143579 (Odious numbers interleaved with Evil numbers); A000069 = Odious numbers, A001969 = Evil numbers). %e A143580 First few terms of A143579 = (1, 0, 2, 3, 4, 5, 7,...), mod 2 = (1, 0, 0, 1, 0, 1, 1,...). %Y A143580 Cf. A010060, A000069, A001969. %K A143580 nonn,new %O A143580 0,1 %A A143580 Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 24 2008 %I A143579 %S A143579 1,0,2,3,4,5,7,6,8,9,11,10,13,12,14,15,16,17,19,18,21,20,22,23,25,24,26, %T A143579 27 %N A143579 Permutation of the natural numbers (0,1,2,3,...): Odious numbers (A000069) interleaved with Evil numbers (A001969). %F A143579 Antidiagonals of an array, Odious numbers (A000069) in row 1 and Evil numbers (A001969) in row 2: 1, 2, 4, 7, 8, 11, 13,... 0, 3, 5, 6, 9, 10, 22,... %Y A143579 Cf. A000069, A001969. %K A143579 nonn,new %O A143579 0,3 %A A143579 Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 24 2008 %I A143564 %S A143564 1,1,7,31,273,1697,16471,116159,1186081,8928193,94017703,736522975, %T A143564 7917810225,63722594657,695248655095,5705316231551,62944217175617, %U A143564 524183926274433,5833380674885959,49141433498848159,550674827214221137 %N A143564 G.f. satisfies: A(x) = 1 + x*A(x)^4/A(-x)^3. %F A143564 G.f. satisfies: A(x) + A(-x) = 1 + (1+x^2)*A(x)*A(-x). %e A143564 G.f. A(x) = 1 + x + 7*x^2 + 31*x^3 + 273*x^4 + 1697*x^5 +... %e A143564 A(x)*A(-x) = 1 + 13*x^2 + 533*x^4 + 32409*x^6 + 2339753*x^8 +... %o A143564 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*A^4/subst(A^3,x,-x));polcoeff(A,n)} %K A143564 nonn,new %O A143564 0,3 %A A143564 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 24 2008 %I A143563 %S A143563 1,1,6,29,242,1554,14476,104061,1024122,7818662,79523444,630256402, %T A143563 6552401972,53271202948,562560238232,4658979320605,49780348483530, %U A143563 418091057783582,4508111500966628,38281314209625862,415790041176520092 %N A143563 G.f. satisfies: A(x) = 1 + x*A(x)^4/A(-x)^2. %F A143563 G.f. satisfies: A(x) + A(-x) = 1 + A(x)*A(-x) + x^2*A(x)^2*A(-x)^2. %e A143563 G.f. A(x) = 1 + x + 6*x^2 + 29*x^3 + 242*x^4 + 1554*x^5 + 14476*x^6 +... %e A143563 A(x)*A(-x) = 1 + 11*x^2 + 462*x^4 + 27907*x^6 + 1982266*x^8 +... %e A143563 A(x)^2*A(-x)^2 = 1 + 22*x^2 + 1045*x^4 + 65978*x^6 + 4791930*x^8 +... %o A143563 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*A^4/subst(A^2,x,-x));polcoeff(A,n)} %K A143563 nonn,new %O A143563 0,3 %A A143563 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 24 2008 %I A143562 %S A143562 1,1,5,17,105,481,3261,16801,119697,656129,4819061,27447601,205776121, %T A143562 1202943457,9152680109,54524185409,419491297313,2534963932417, %U A143562 19673179986661,120224135048273,939543098579081,5793676726569697 %N A143562 G.f. satisfies: A(x) = 1 + x*A(x)^3/A(-x)^2. %F A143562 G.f. satisfies: A(x) + A(-x) = 1 + (1+x^2)*A(x)*A(-x). %e A143562 G.f. A(x) = 1 + x + 5*x^2 + 17*x^3 + 105*x^4 + 481*x^5 + 3261*x^6 +... %e A143562 A(x)*A(-x) = 1 + 9*x^2 + 201*x^4 + 6321*x^6 + 233073*x^8 +... %o A143562 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*A^3/subst(A^2,x,-x));polcoeff(A,n)} %K A143562 nonn,new %O A143562 0,3 %A A143562 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 24 2008 %I A143561 %S A143561 1,2,9,24,88,280,1064,3672,14456,52184,210504,782232,3210904,12176792, %T A143561 50638440,194956248,818961080,3189915224,13508052104,53105011480, %U A143561 226355549400,896636646936,3842662060200,15317408281944,65946510374136 %N A143561 G.f. satisfies: A(x) = ( 1 + x*A(x)/A(-x) )^2. %F A143561 G.f. satisfies: (1+x^2)^2 - 2*(1+x^2)*G(x) + (1+x)*G(x)^2 - x*G(x)^3 = 0 where G(x)^2 = A(x) and G(x) = 1 + x*A(x)/A(-x) is the g.f. of A143555. %e A143561 G.f.: A(x) = 1 + 2*x + 9*x^2 + 24*x^3 + 88*x^4 + 280*x^5 + 1064*x^6 +... %e A143561 A(x)/A(-x) = 1 + 4*x + 8*x^2 + 28*x^3 + 80*x^4 + 308*x^5 + 984*x^6 +... %o A143561 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,B=A/subst(A,x,-x);A=(1+x*B)^2);polcoeff(A,n)} %Y A143561 Cf. A143555. %K A143561 nonn,new %O A143561 0,2 %A A143561 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 24 2008 %I A143560 %S A143560 1,1,3,6,16,38,110,276,818,2158,6528,17766,54622,151852,472674,1334886, %T A143560 4195328,11992486,37981982,109622228,349384626,1016304750,3256170672, %U A143560 9533400198,30680043630,90318157804,291763419458,862944630022 %N A143560 G.f. satisfies: A(x) = 1 + x*A(x)/A(-x) + x^2*A(x)^2/A(-x)^2. %e A143560 A(x) = 1 + x + 3*x^2 + 6*x^3 + 16*x^4 + 38*x^5 + 110*x^6 + 276*x^7 +... %e A143560 A(x)/A(-x) = 1 + 2*x + 2*x^2 + 8*x^3 + 14*x^4 + 46*x^5 + 96*x^6 +... %e A143560 A(x)^2/A(-x)^2 = 1 + 4*x + 8*x^2 + 24*x^3 + 64*x^4 + 180*x^5 +... %o A143560 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,B=A/subst(A,x,-x);A=1+x*B+x^2*B^2);polcoeff(A,n)} %K A143560 nonn,new %O A143560 0,3 %A A143560 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 24 2008 %I A143578 %S A143578 1,2,3,5,7,11,13,15,17,19,23,29,31,35,37,41,43,47 %N A143578 A positive integer n is included if (j+n/j) divides (k+n/k) for every divisor k of n, where j is the largest divisor of n that is <= sqrt(n). %C A143578 This sequence trivially contains all the primes. %e A143578 The divisors of 35 are 1,5,7,35. The sum of the two middle divisors is 5+7 = 12. 12 divides 7 + 35/7 = 5+35/5 = 12, of course. And 12 divides 1 + 35/1 = 35 +35/35 = 36. So 35 is in the sequence. %Y A143578 Cf. A063655. %K A143578 more,nonn,new %O A143578 1,2 %A A143578 Leroy Quet (qq-quet(AT)mindspring.com), Aug 24 2008 %I A143575 %S A143575 0,1,4,5,9,10,13,16,17,20,26,29,34,36,37,40,41,45,49,52,53,58,61,64,68, %T A143575 73,74,80,81,82,89,90,97,101,104,106,109,113,116,117,121,122,136,137, %U A143575 144,146,148,149,153,157,160,164,173,178,180,181,193,194,196,197,202 %N A143575 Numbers m such that A143574(m) = m. %C A143575 A000161(a(n)) = 1; %C A143575 A000548 is a subsequence. %H A143575 R. Zumkeller, Table of n, a(n) for n = 1..10000 %K A143575 nonn,new %O A143575 1,3 %A A143575 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 24 2008 %I A143574 %S A143574 0,1,1,0,4,5,0,0,4,9,10,0,0,13,0,0,16,17,9,0,20,0,0,0,0,50,26,0,0,29,0, %T A143574 0,16,0,34,0,36,37,0,0,40,41,0,0,0,45,0,0,0,49,75,0,52,53,0,0,0,0,58,0, %U A143574 0,61,0,0,64,130,0,0,68,0,0,0,36,73,74,0,0,0,0,0,80,81,82,0,0,170,0,0,0 %N A143574 Sum of all distinct squares occurring when partitioning n into two squares. %C A143574 For n > 0: a(n) = 0 iff A000161(n) = 0: a(A022544(n)) = 0; %C A143574 A143575 gives numbers m such that a(m) = m. %H A143574 R. Zumkeller, Table of n, a(n) for n = 0..10000 %e A143574 A000161(25)=#{5^2+0^2,4^2+3^2}=2: a(25)=25+0+16+9=50; %e A143574 A000161(26)=#{5^2+1^2}=1: a(16)=25+1=26; %e A143574 A000161(49)=#{7^2+0^2}=1: a(49)=49+0=49; %e A143574 A000161(50)=#{7^2+1^2,5^2+5^2}=2: a(50)=49+1+25=75; %e A143574 A000161(2600)=#{50^2+10^2,46^2+22^2,38^2+34^2}=3: %e A143574 a(2600)=2500+100+2116+484+1444+1156=7800; %e A143574 A000161(2601)=#{51^2+0^2,45^2+24^2}=2: %e A143574 a(2601)=2601+0+12025+576=5202; %e A143574 A000161(2602)=#{51^2+1^2}=1: a(26002)=2601+1=2602. %K A143574 nonn,new %O A143574 0,5 %A A143574 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 24 2008 %I A143559 %S A143559 1,1,12,72,1012,9552,148764,1609496,26398020,305821344,5174354988, %T A143559 62479377384,1079265357204,13399747245040,234917433809724, %U A143559 2975608178304696,52748683164797668,678307369324850496 %N A143559 G.f. satisfies: A(x) = 1 + x*A(x)^6/A(-x)^6. %F A143559 G.f. satisfies: A(x) = 1 + x^2/(1 - A(-x)). %F A143559 G.f. satisfies: (A(x) - 1)^5 = ( 1 - (1+x^2)/A(x) )^6/x = x^5*A(x)^30/A(-x)^30. %e A143559 G.f. A(x) = 1 + x + 12*x^2 + 72*x^3 + 1012*x^4 + 9552*x^5 + 148764*x^6 +... %e A143559 A(x)/A(-x) = 1 + 2*x + 2*x^2 + 122*x^3 + 242*x^4 + 16002*x^5 + 38962*x^6 +... %e A143559 A(x)^5/A(-x)^5 = 1 + 10*x + 50*x^2 + 770*x^3 + 6450*x^4 + 109802*x^5 +... %e A143559 where 1 - (1+x^2)/A(x) = x*A(x)^5/A(-x)^5. %o A143559 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*A^6/subst(A^6,x,-x));polcoeff(A,n)} %Y A143559 Cf. A143555, A143556, A143557, A143558. %K A143559 nonn,new %O A143559 0,3 %A A143559 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 24 2008 %I A143558 %S A143558 1,1,10,50,570,4450,56202,501970,6676410,63799490,875391370,8715058802, %T A143558 122088479930,1249437863970,17764858122250,185445650940690, %U A143558 2666213981716282,28252030821781890,409717783914784010 %N A143558 G.f. satisfies: A(x) = 1 + x*A(x)^5/A(-x)^5. %F A143558 G.f. satisfies: A(x) = 1 + x^2/(1 - A(-x)). %F A143558 G.f. satisfies: (A(x) - 1)^4 = ( 1 - (1+x^2)/A(x) )^5/x = x^4*A(x)^20/A(-x)^20. %e A143558 G.f. A(x) = 1 + x + 10*x^2 + 50*x^3 + 570*x^4 + 4450*x^5 + 56202*x^6 +... %e A143558 A(x)/A(-x) = 1 + 2*x + 2*x^2 + 82*x^3 + 162*x^4 + 7202*x^5 + 17442*x^6 +... %e A143558 A(x)^4/A(-x)^4 = 1 + 8*x + 32*x^2 + 408*x^3 + 2752*x^4 + 38760*x^5 +... %e A143558 where 1 - (1+x^2)/A(x) = x*A(x)^4/A(-x)^4. %o A143558 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*A^5/subst(A^5,x,-x));polcoeff(A,n)} %Y A143558 Cf. A143555, A143556, A143557, A143559. %K A143558 nonn,new %O A143558 0,3 %A A143558 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 24 2008 %I A143557 %S A143557 1,1,8,32,280,1728,16744,117856,1202552,9044352,95203784,745451168, %T A143557 8011827928,64459117632,703166465320,5769038826208,63639465830712, %U A143557 529889242505984,5896324892061576,49665617425122592,556508207889107096 %N A143557 G.f. satisfies: A(x) = 1 + x*A(x)^4/A(-x)^4. %F A143557 G.f. satisfies: A(x) = 1 + x^2/(1 - A(-x)). %F A143557 G.f. satisfies: (A(x) - 1)^3 = ( 1 - (1+x^2)/A(x) )^4/x = x^3*A(x)^12/A(-x)^12. %e A143557 G.f. A(x) = 1 + x + 8*x^2 + 32*x^3 + 280*x^4 + 1728*x^5 + 16744*x^6 +... %e A143557 A(x)/A(-x) = 1 + 2*x + 2*x^2 + 50*x^3 + 98*x^4 + 2658*x^5 + 6370*x^6 +... %e A143557 A(x)^3/A(-x)^3 = 1 + 6*x + 18*x^2 + 182*x^3 + 930*x^4 + 10374*x^5 +... %e A143557 where 1 - (1+x^2)/A(x) = x*A(x)^3/A(-x)^3. %o A143557 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*A^4/subst(A^4,x,-x));polcoeff(A,n)} %Y A143557 Cf. A143555, A143556, A143558, A143559. %K A143557 nonn,new %O A143557 0,3 %A A143557 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 24 2008 %I A143556 %S A143556 1,1,6,18,110,498,3366,17282,122958,672930,4938758,28103730,210595182, %T A143556 1230391058,9358456230,55727128866,428643977422,2589488117826, %U A143556 20092671283974,122759098980690,959216278565742,5913900861617970 %N A143556 G.f. satisfies: A(x) = 1 + x*A(x)^3/A(-x)^3. %F A143556 G.f. satisfies: A(x) = 1 + x^2/(1 - A(-x)). %F A143556 G.f. satisfies: (A(x) - 1)^2 = ( 1 - (1+x^2)/A(x) )^3/x = x^2*A(x)^6/A(-x)^6. %e A143556 G.f. A(x) = 1 + x + 6*x^2 + 18*x^3 + 110*x^4 + 498*x^5 + 3366*x^6 +... %e A143556 A(x)/A(-x) = 1 + 2*x + 2*x^2 + 26*x^3 + 50*x^4 + 706*x^5 + 1650*x^6 +... %e A143556 A(x)^2/A(-x)^2 = 1 + 4*x + 8*x^2 + 60*x^3 + 208*x^4 + 1716*x^5 +... %e A143556 where 1 - (1+x^2)/A(x) = x*A(x)^2/A(-x)^2. %o A143556 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*A^3/subst(A^3,x,-x));polcoeff(A,n)} %Y A143556 Cf. A143555, A143557, A143558, A143559. %K A143556 nonn,new %O A143556 0,3 %A A143556 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 24 2008 %I A143555 %S A143555 1,1,4,8,28,80,308,984,3980,13472,56164,197032,838396,3013872,13015188, %T A143555 47624568,207971436,771336512,3397886660,12736715592,56502898140, %U A143555 213618833808,953139545076,3629043226392,16270547827020,62317467147744 %N A143555 G.f. satisfies: A(x) = 1 + x*A(x)^2/A(-x)^2. %F A143555 G.f. satisfies: A(x) = 1 + x^2/(1 - A(-x)). %F A143555 G.f. satisfies: A(x) = 1 + ( 1 - (1+x^2)/A(x) )^2/x. %F A143555 G.f. satisfies: (1+x^2)^2 - 2*(1+x^2)*A(x) + (1+x)*A(x)^2 - x*A(x)^3 = 0. %e A143555 G.f. A(x) = 1 + x + 4*x^2 + 8*x^3 + 28*x^4 + 80*x^5 + 308*x^6 +... %e A143555 A(x)/A(-x) = 1 + 2*x + 2*x^2 + 10*x^3 + 18*x^4 + 98*x^5 + 210*x^6 +... %e A143555 where 1 - (1+x^2)/A(x) = x*A(x)/A(-x). %o A143555 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*A^2/subst(A^2,x,-x));polcoeff(A,n)} %Y A143555 Cf. A143554, A143556, A143557, A143558, A143559. %K A143555 nonn,new %O A143555 0,3 %A A143555 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 24 2008 %I A141631 %S A141631 2,7,18,35,58,87,122,163,210,263,322,387,458,535,618,707,802,903,1010 %N A141631 First bisection of yesterday 2, 5, 7, 14, 18, 29, 35 . Also: a(n)=(n-1)^2+n(n+2)+(n+1)^2 . %F A141631 Differences: 5, 11, 17, 23 = A016969. Second differences: 6, 6, 6 = A010722 . Third differences: 0, 0, 0 = A000004 . %K A141631 nonn,uned,new %O A141631 1,1 %A A141631 Paul Curtz (bpcrtz(AT)free.fr), Aug 28 2008 %I A141630 %S A141630 1,1,3,5,13,26,66,152,366,907,2225,6864 %N A141630 Number of constitutional isomers of alkylcyclobutadienes. %D A141630 Ching-Wan Lam, "Enumeration of isomers of alkylcyclobutadienes by means of alkyl 1,1,1-triradicals", J. Math. Chem., vol. 31 (2002), pp. 333 - 337. See Table 1 on page 336. %e A141630 If n=15 then the number of constitutional isomers of alkylcyclobutadienes is 6864. %K A141630 nonn,new %O A141630 4,3 %A A141630 Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Aug 28 2008 %I A139699 %S A139699 1,5,3,22,10,186 %N A139699 Positive integers x,y such that Prime(x)Prime(x+y) = y^2 + 1, ordered by increasing x+y. %H A139699 Sebastian Martin Ruiz Prime Pages: Home Page %e A139699 Prime(1)*Prime(1+5)=2*13=26=5^2+1 %K A139699 nonn,new %O A139699 1,2 %A A139699 Sebastian Martin-Ruiz (s_m_ruiz(AT)yahoo.es), Aug 25 2008 %I A134345 %S A134345 1,1,1,2,2,3,4,5,6,7,9,11,13,16,19,23,27,32,38,44,52,61,71,82,95,109, %T A134345 126,144,165,189,215,245,278,316,358,405,458,516,581,654,734,824,923, %U A134345 1033,1155,1289,1438,1602,1783,1982,2202,2444,2710,3002,3323,3675,4061 %N A134345 Number of partitions into odd squarefree parts. %C A134345 Also number of partitions into parts m such that 2*m is squarefree. %H A134345 fxtbook (Joerg Arndt) %F A134345 1/prod(n=1,infinity, 1-moebius(2*n-1)^2*x^(2*n-1)) 1/prod(n=1,infinity, 1-moebius(2*n)^2*x^(n)) %o A134345 (PARI) Vec( 1/prod(n=1,1000, 1-moebius(2*n-1)^2*x^(2*n-1) ) ) %K A134345 nonn,new %O A134345 1,4 %A A134345 Joerg Arndt (arndt(AT)jjj.de), Aug 27 2008 %I A134337 %S A134337 1,1,0,1,1,1,1,1,2,1,1,2,2,2,2,3,4,3,4,5,5,6,6,7,8,7,8,9,9,11,10,12,14, %T A134337 14,16,17,20,21,21,25,27,27,29,31,35,35,36,42,44,45,49,55,59,61,66,74, %U A134337 77,81,87,93,99,102,110,117,123,131,138,148,159,167,178,190,204,215,225 %N A134337 Number of partitions into distinct odd squarefree parts. %C A134337 Also number of partitions into distinct parts m such that 2*m is squarefree %H A134337 fxtbook (Joerg Arndt) %F A134337 prod(n=1,infinity, 1+moebius(2*n-1)^2*x^(2*n-1) ) ) prod(n=1,infinity, 1+moebius(2*n)^2*x^(n) ) ) %o A134337 (PARI) Vec( prod(n=1,1000, 1+moebius(2*n-1)^2*x^(2*n-1) ) ) /* also Vec( prod(n=1,1000, 1+moebius(2*n)^2*x^(n) ) ) */ %K A134337 nonn,new %O A134337 0,9 %A A134337 Joerg Arndt (arndt(AT)jjj.de), Aug 27 2008 %I A134307 %S A134307 11,29,37,43,59,71,79,97,103,109,113,127,131,137,151,163,181,191,197, %T A134307 199,211,223,229,233,241,257,263,269,281,283,293,307,313,331,347,349, %U A134307 353,359,367,373,379,397,401,419,421,433,439,449,461,463,487,499,509 %N A134307 Primes such that A^(p-1)==1 (mod P^2) for some A where 1Index entries for sequences related to rooted trees %F A134306 See program. %p A134306 a:= proc(n) local B,z; B:= proc (x,y,d) if d>=1 then x+B(x^2+2*x*y, x,d-1) else x fi end; coeff(B(z,0,6), z,n) end: seq (a(n), n=0..64); %Y A134306 Cf. A006265, A036662. %K A134306 fini,nonn,new %O A134306 0,4 %A A134306 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 27 2008 %I A134261 %S A134261 0,1,2,9,68,735,10332,178276,3639680,85750461,2289322710,68298539441, %T A134261 2251768422840,81301875813340,3190478732975744,135209859332836905, %U A134261 6154229137942791184,299422872446882413387,15507211446546229257948 %N A134261 E.g.f. satisfies: A(x) = x*(sinh(exp(A(n-1))-1)+1). %p A134261 A:= proc(n) option remember; if n=0 then 0 else convert (series (x* (sinh (exp(A(n-1))-1)+1), x=0, n+1), polynom) fi end: a:= n-> coeff (A(n), x, n)*n!: seq (a(n), n=0..22); %K A134261 nonn,new %O A134261 0,3 %A A134261 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 27 2008 %I A134200 %S A134200 0,1,2,9,68,725,9942,166453,3290632,75017097,1937420010,55906879809, %T A134200 1782695466444,62247810769053,2362246665531326,96806321000599725, %U A134200 4260677055123222544,200440759819510706321,10037364633737549049042 %N A134200 E.g.f. satisfies: A(x) = x*(exp(sinh(A(n-1)))). %p A134200 A:= proc(n) option remember; if n=0 then 0 else convert (series (x* (exp (sinh(A(n-1)))), x=0, n+1), polynom) fi end: a:= n-> coeff (A(n), x, n)*n!: seq (a(n), n=0..22); %K A134200 nonn,new %O A134200 0,3 %A A134200 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 27 2008 %I A133984 %S A133984 0,1,2,9,72,845,12972,244741,5468176,141111693,4129615540,135127313101, %T A133984 4888457921688,193733261456605,8346805786382364,388432439875807125, %U A133984 19417284993350451232,1037672210204182995277,59035412382992193993732 %N A133984 E.g.f. satisfies: A(x) = x*(tan(exp(A(x))-1)+1). %p A133984 A:= proc(n) option remember; if n=0 then 0 else convert (series (x* (tan (exp(A(n-1))-1)+1), x=0, n+1), polynom) fi end: a:= n-> coeff (A(n), x, n)*n!: seq (a(n), n=0..22); %K A133984 nonn,new %O A133984 0,3 %A A133984 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 27 2008 %I A133941 %S A133941 0,1,2,9,72,825,12192,220353,4708480,116116497,3245839360,101415497689, %T A133941 3502465714176,132486192976137,5447446920323072,241907419042038225, %U A133941 11538444129924055040,588321821566662253729,31932991994214557417472 %N A133941 E.g.f. satisfies: A(x) = x*(exp(tan(A(x)))). %p A133941 A:= proc(n) option remember; if n=0 then 0 else convert (series (x* (exp (tan(A(n-1)))), x=0, n+1), polynom) fi end: a:= n-> coeff (A(n), x, n)*n!: seq (a(n), n=0..22); %K A133941 nonn,new %O A133941 0,3 %A A133941 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 27 2008 %I A133939 %S A133939 0,1,2,6,40,440,5952,97104,1888640,42480000,1082119680,30814080000, %T A133939 970187827200,33461288899584,1254539018571776,50803163905751040, %U A133939 2209862882578300928,102761280728930287616,5087062588875762696192 %N A133939 E.g.f. satisfies: A(x) = x*(tan(tan(A(x)))+1). %p A133939 A:= proc(n) option remember; if n=0 then 0 else convert (series (x* (tan (tan(A(n-1)))+1), x=0, n+1), polynom) fi end: a:= n-> coeff (A(n), x, n)*n!: seq (a(n), n=0..23); %K A133939 nonn,new %O A133939 0,3 %A A133939 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 27 2008 %I A133892 %S A133892 0,1,2,6,36,360,4542,68544,1226792,25441920,598142170,15713984000, %T A133892 456391238028,14521095333888,502259604707798,18763725111828480, %U A133892 752970270575818192,32301914469949407232,1475208429063535282482 %N A133892 E.g.f. satisfies: A(x) = x*(tan(sinh(A(x)))+1). %p A133892 A:= proc(n) option remember; if n=0 then 0 else convert (series (x* (tan (sinh(A(n-1)))+1), x=0, n+1), polynom) fi end: a:= n-> coeff (A(n), x, n)*n!: seq (a(n), n=0..23); %K A133892 nonn,new %O A133892 0,3 %A A133892 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 27 2008 %I A133822 %S A133822 0,1,2,6,36,360,4542,68544,1226344,25409664,596628250,15651680000, %T A133822 453879958092,14417575231488,497825878940054,18565202648401920, %U A133822 743653004987969360,31843195958676979712,1451524546915205994162 %N A133822 E.g.f. satisfies: A(x) = x*(sinh(tan(A(x)))+1). %p A133822 A:= proc(n) option remember; if n=0 then 0 else convert (series (x* (sinh (tan(A(n-1)))+1), x=0, n+1), polynom) fi end: a:= n-> coeff (A(n), x, n)*n!: seq (a(n), n=0..23); %K A133822 nonn,new %O A133822 0,3 %A A133822 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 27 2008 %I A133596 %S A133596 0,1,2,6,32,280,3192,43344,690496,12726144,266222880,6222163200, %T A133596 160658284800,4542751030272,139616399952512,4634016219678720, %U A133596 165191949394008064,6294553527003086848,255316547059075256832 %N A133596 E.g.f. satisfies: A(x) = x*(sinh(sinh(A(x)))+1). %p A133596 A:= proc(n) option remember; if n=0 then 0 else convert (series (x* (sinh (sinh(A(n-1)))+1), x=0, n+1), polynom) fi end: a:= n-> coeff (A(n), x, n)*n!: seq (a(n), n=0..23); %K A133596 nonn,new %O A133596 0,3 %A A133596 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 27 2008 %I A133553 %S A133553 0,1,0,3,12,120,1290,17409,277592,5083659,105675030,2452220144, %T A133553 62891640900,1766131052829,53900956145218,1776400037307315, %U A133553 62874491729108656,2378684861565934468,95790461019732936558 %N A133553 E.g.f. satisfies: A(x) = x*(sec(exp(A(x))-1)). %p A133553 A:= proc(n) option remember; if n=0 then 0 else convert (series (x* (sec (exp(A(n-1))-1)), x=0, n+1), polynom) fi end: a:= n-> coeff (A(n), x, n)*n!: seq (a(n), n=0..24); %K A133553 nonn,new %O A133553 0,4 %A A133553 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 27 2008 %I A133359 %S A133359 0,1,0,3,12,100,1050,12649,185752,3112407,59052390,1252912584, %T A133359 29341892580,752441547741,20966217326418,630757511101995, %U A133359 20377626191365936,703606826009437384,25858057389119292222 %N A133359 E.g.f. satisfies: A(x) = x*(cosh(exp(A(x))-1)). %p A133359 A:= proc(n) option remember; if n=0 then 0 else convert (series (x* (cosh (exp(A(n-1))-1)), x=0, n+1), polynom) fi end: a:= n-> coeff (A(n), x, n)*n!: seq (a(n), n=0..24); %K A133359 nonn,new %O A133359 0,4 %A A133359 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 27 2008 %I A133280 %S A133280 0,1,3,4,6,8,9,11,13,15,16,18,20,22,24,25,27,29,31,33,35,36,38,40,42,44, %T A133280 46,48,49,51,53,55,57,59,61,63,64,66,68,70,72,74,76,78,80,81,83,85,87, %U A133280 89,91,93,95,97,99,100,102,104,106,108,110,112,114,116,118,120 %N A133280 Triangle formed by: 1 even, 2 odd, 3 even, 4 odd... starting with zero. %C A133280 This sequence is related to the Connell sequence (A001614). %C A133280 First member of every row is a square (A000290). %e A133280 Triangle begins: %e A133280 0 %e A133280 1, 3 %e A133280 4, 6, 8 %e A133280 9, 11, 13, 15 %e A133280 16, 18, 20, 22, 24 %e A133280 25, 27, 29, 31, 33, 35 %Y A133280 Cf. A000290, A001614, A005563. %K A133280 easy,nonn,tabl,new %O A133280 0,3 %A A133280 Omar E. Pol (info(AT)polprimos.com), Aug 27 2008 %I A133279 %S A133279 1,0,1,1,7,21,234 %N A133279 Number of unlabeled mating graphs with n nodes and a degenerate adjacency matrix. %C A133279 Mating graphs are graphs where no two nodes have the same set of neighbors. %C A133279 Graphs with an invertible adjacency matrix are mating graphs. %C A133279 a(n) = A004110(n) - A109717(n). %t A133279 k = {}; For[i = 1, i < 8, i++, lg = ListGraphs[i] ; len = Length[lg]; k = Append[k, Length[Select[Range[len], Det[ToAdjacencyMatrix[lg[[ # ]]]] == 0 && Length[Union[ToAdjacencyMatrix[lg[[ # ]]]]] == i &]]]]; k %Y A133279 Cf. A004110, A109717. %K A133279 more,nonn,new %O A133279 1,5 %A A133279 Tanya Khovanova (tanyakh(AT)yahoo.com), Aug 27 2008 %I A133206 %S A133206 1,1,3,7,25,99,690 %N A133206 Number of unlabeled graphs with n nodes and a degenerate adjacency matrix. %C A133206 a(n) = A000088(n) - A109717(n) %t A133206 k = {}; For[i = 1, i < 8, i++, lg = ListGraphs[i] ; len = Length[lg]; k = Append[k, Length[Select[Range[len], Det[ToAdjacencyMatrix[lg[[ # ]]]] == 0 &]]]]; k %Y A133206 Cf. A000088, A109717. %K A133206 more,nonn,new %O A133206 1,3 %A A133206 Tanya Khovanova (tanyakh(AT)yahoo.com), Aug 27 2008 %I A133203 %S A133203 0,1,3,31,67,111,163,223,291,367,451,543,643,751,867,991,1123,1263,1411, %T A133203 1567,1731,1903,2083,2271,2467,2671,2883,3103,3331,3567,3811 %N A133203 a(n)=a(n-1)+8*n+4. %C A133203 Apart from {0,1} at the start, 15 of the first 30 are primes. %e A133203 fz(1)=0; fz(2)=1; fz(3)=3; %e A133203 for k=4:n %e A133203 fz(k)=fz(k-1)+8*k+4; %e A133203 end %e A133203 y=fz(n); %o A133203 MatLab program: function y=fib(n) %K A133203 nonn,new %O A133203 1,3 %A A133203 Matt Wynne (mattwyn(AT)verizon.net), Aug 27 2008 %I A133146 %S A133146 2,5,7,14,18,29,35,50,58,77,87,110,122,149,163,194,210,245,263,302 %N A133146 (Based on A120070). Antidiagonals of today triangle in 2, 5, -3, 10, -3, 5 . %C A133146 Second bisection: 5, 14, 29, 50, 77 = A005918(n+1). %F A133146 a(n) differences are 3, 2, 7, 4, 11 = A059029(n+1). %K A133146 nonn,uned,new %O A133146 0,1 %A A133146 Paul Curtz (bpcrtz(AT)free.fr), Aug 27 2008 %I A133128 %S A133128 2,5,3,10,3,5,17,3,5,7,26,3,5,7,9,37,3,5,7,9,11,50,3,5,7,9,11,13,65,3,5, %T A133128 7,9,11,13,15 %V A133128 2,5,-3,10,-3,-5,17,-3,-5,-7,26,-3,-5,-7,-9,37,-3,-5,-7,-9,-11,50,-3,-5,-7,-9,-11,-13, %W A133128 65,-3,-5,-7,-9,-11,-13,-15 %N A133128 Based on A120070. For a triangle: 2, A141620. On-line. %F A133128 Triangle: 2; 5, -3; 10, -3, -5; 17, -3, -5, -7; First column: A002522(n+1). Rows sum: A007395 . Diagonal: 2, -A005408(n+1) . %K A133128 sign,uned,new %O A133128 0,1 %A A133128 Paul Curtz (bpcrtz(AT)free.fr), Aug 27 2008 %I A132993 %S A132993 1,2,2,3,4,3,5,6,6,5,7,10,9,10,7,11,14,15,15,14,11,15,22,21,25,21,22,15, %T A132993 22,30,33,35,35,33,30,22,30,44,45,55,49,55,45,44,30,42,60,66,75,77,77, %U A132993 75,66,60,42,56,84,90,110,105,121,105,110,90,84,56 %N A132993 P partitions (A000041) weight symmetrical triangle of coefficients: t(n,m)=PartitionsP[n - m + 1]*PartitionsP[m + 1]. %C A132993 Row sums are: %C A132993 {1, 4, 10, 22, 43, 80, 141, 240, 397, 640, 1011}. %D A132993 Weisstein, Eric W. "Partition." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Partition.html %F A132993 t(n,m)=PartitionsP[n - m + 1]*PartitionsP[m + 1]. %e A132993 {1}, %e A132993 {2, 2}, %e A132993 {3, 4, 3}, %e A132993 {5, 6, 6, 5}, %e A132993 {7, 10, 9, 10, 7}, %e A132993 {11, 14, 15, 15, 14, 11}, %e A132993 {15, 22, 21, 25, 21, 22, 15}, %e A132993 {22, 30, 33, 35, 35, 33, 30, 22}, %e A132993 {30, 44, 45, 55, 49, 55, 45, 44, 30}, %e A132993 {42, 60, 66, 75, 77, 77, 75, 66, 60, 42}, %e A132993 {56, 84, 90, 110, 105, 121, 105, 110, 90, 84, 56} %t A132993 << DiscreteMath`Combinatorica`; << DiscreteMath`IntegerPartitions`; Clear[t, n, m]; t[n_, m_] = PartitionsP[n - m + 1]*PartitionsP[m + 1]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%] %Y A132993 Cf. A000041. %K A132993 nonn,uned,tabl,new %O A132993 1,2 %A A132993 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 27 2008 %I A132992 %S A132992 32970,180180,273000,633570,879690,991620,1189650,2219490,3229380, %T A132992 4111170,4515630,7384440,7392630,7398930,7431270,9022440,9861390 %N A132992 Averages of Twin Primes if : Sum of Lower, Average and Upper part of Twin Primes are Average of the other Twin Primes, and remains true through 3 iterations. %e A132992 32970 -> 98910 -> 296730 Twin Prime Averages %t A132992 TwinPrimeAverageQ[n_]:=If[PrimeQ[n-1]&&PrimeQ[n+1],True,False](*TwinPrimeAverageQ*)lst={};Do[If[TwinPrimeAverageQ[n],If[TwinPrimeAverageQ[3*n],If[TwinPrimeAverageQ[9*n],(*Print[n];*)AppendTo[lst,n]]]],{n,7!,3*10!}];lst %K A132992 nonn,new %O A132992 1,1 %A A132992 Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 26 2008 %I A132929 %S A132929 4,6,60,270,1950,3000,6360,11490,11550,14550,18540,19890,21840,31080, %T A132929 32910,32970,33330,33600,42570,42840,50460,53550,58110,68880,70200, %U A132929 74610,79230,80910,93810,96330,98910,104310,109140,114600,121020,125790 %N A132929 Averages of Twin Primes if : Sum of Lower, Average and Upper part of Twin Primes are Averages of the other Twin Primes. %e A132929 3+4+5=12 -> 11,13, %e A132929 5+6+7=18 -> 17,19, %e A132929 59+60+61=180 -> 179,181 %t A132929 TwinPrimeAverageQ[n_]:=If[PrimeQ[n-1]&&PrimeQ[n+1],True,False](*TwinPrimeAverageQ*) lst={};Do[If[TwinPrimeAverageQ[n],If[TwinPrimeAverageQ[3*n],(*Print[n];*)AppendTo[lst,n]]],{n,9!}];lst %K A132929 nonn,new %O A132929 1,1 %A A132929 Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 26 2008 %I A132635 %S A132635 0,1,4,6,8,11,13,17,20,24,27,32,36,41,46,50,56,63,68,74,80,87,94,101, %T A132635 107,116,124,131,139,148,156 %N A132635 Number of either prime-like or {0,1]-like levels in each square n: a(n)=Sum[If[m == 0 || m == 1, 1, If[PrimeQ[m], 1, 0]], {m, 0, n^2 - 1}]. %C A132635 Number of p-adic cyclotomic type levels in A_n-1 or SU(n). %F A132635 a(n)=Sum[If[m == 0 || m == 1, 1, If[PrimeQ[m], 1, 0]], {m, 0, n^2 - 1}]. %e A132635 SU(5) at n^2-1=24 is exactly inside SU(9) at n^2-1=80. %t A132635 Table[Sum[If[m == 0 || m == 1, 1, If[PrimeQ[m], 1, 0]], {m, 0,n^2 - 1}], {n, 0, 30}] %K A132635 nonn,new %O A132635 1,3 %A A132635 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 26 2008 %I A132404 %S A132404 3,20,60,204,1,420,660,2040 %V A132404 3,20,60,204,-1,420,660,2040 %N A132404 Smallest short legs 'A' of exactly n primitive Pythagorean triangles, or -1 if no such shortest leg exists. %e A132404 1, 3.4.5 %e A132404 2, 20.21.29, 20.99.101 %e A132404 3, 60.91.109, 60.221.229, 60.899.901 %e A132404 4, 204.253.325, 204.1147.1165, 204.2597.2605, 204.10403.10405 %e A132404 5, -1 -- No numbers can represent short legs 'A' of exactly 5 primitive Pythagorean triangles. %e A132404 6, 420.851.949, 420.1189.1261, 420.1739.1789, 420.4891.4909, 420.11021.11029, 420.44099.44101 %e A132404 7, 660.779.1021, 660.989.1189, 660.2989.3061, 660.4331.4381, 660.12091.12109, 660.27221.27229, 660.108899.108901 %t A132404 PyphagoreanAs[a_]:=(q={};k=0;If[a>=8,r=4,r=1];Do[y=(a^2+b^2)^0.5;c=IntegerPart[y];If[c==y,p=0;If[GCD[a,b,c]==1,AppendTo[q,a.b.c];k++ ]],{b,a+1,a^2/r}];PrependTo[q,k];q)lst={};x=0;Do[w=PyphagoreanAs[n][[1]];If[w>x,Print[Date[],"A=",n,",w=",w];AppendTo[lst,n];x=w],{n,7!}];lst %K A132404 sign,new %O A132404 1,1 %A A132404 Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 26 2008 %I A132390 %S A132390 3,6,24,76,288,1072,4224,16576,66048,262912 %N A132390 Number of binary pattern classes in the (2,n)-rectangular grid; two patterns are in same class if one of them can be obtained by reflexion or rotation of the other one. %C A132390 A005418 is the solution for the problem in the (1,n)-rectangular grid %Y A132390 Cf. A005418, A034851. %K A132390 hard,more,nonn,new %O A132390 1,1 %A A132390 Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Aug 26 2008 %I A132169 %S A132169 2,3,6,4,8,5,12,10,6,15,12,7,20,18,14,8,24,21,16,9,30,28,24,18,10 %N A132169 Based on A120070. A141616/4. %K A132169 nonn,uned,new %O A132169 0,1 %A A132169 Paul Curtz (bpcrtz(AT)free.fr), Aug 26 2008 %I A132084 %S A132084 1,3,30,42,30,66,2730,6,510,798,330,138,2730,6,870,14322,510,6,1919190, %T A132084 6,13530,1806,690,282,46410,66,1590,798,870,354 %N A132084 Denominators of Bernoulli twin numbers C(n). First bisection: A051717(2n). %C A132084 a(2n)+a(2n+1)= 4, 72, 96, 2736, 1308, 468 , multiples of 4. %K A132084 nonn,uned,new %O A132084 0,2 %A A132084 Paul Curtz (bpcrtz(AT)free.fr), Aug 26 2008 %I A131952 %S A131952 2047,8388607,1082401,3277,536870911,8727391,4033,137438953471,9588151 %N A131952 Maximal overpseudoprimes a(n) to base 2 with the multiplicative order of 2 mod a(n) equals to A143584(n). %C A131952 Or composite terms of A064078. %Y A131952 Cf. A143584 A064078 A141232 A122929 A141629 A002326. %K A131952 nonn,new %O A131952 1,1 %A A131952 Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 26 2008 %I A131386 %S A131386 1,2,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0 %V A131386 1,-2,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0 %N A131386 We start from a generalized Diophantine Equation : Z^n=X_1^{n_1}+...X_i^{n_i} n_j , n , X_j, Z are positive integers, X_j, Z are coprime. For ,i=2, n_j=n it is Fermat equation. For i=2 , it is Fermat-Catalan (or Beal). After a little change of the data, we define the following sequences (articles published by the Asian Journal of Algebra, copyright) x_i= rac{x^{2^{i-1}}}{x^{2^{i-1}}-y^{2^{i-1}}}(x-y) y_i= rac{y^{2^{i-1}}}{x^{2^{i-1}}-y^{2^{i-1}}}(x-y) z_i=x_i+y_i u_i= rac{x_iy_i}{x_i+y_i} The coefficients of z_i in function of z_{i-1} and u_{i-1} beginning from i=3 . The sequence is, then 1, -2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0,… (We proved that x_i-y_i=x-y=0 ). %C A131386 This sequence is generated by a generalized Diophantine Equation. It has no formula, but it seems that a(2k+1)=1 for all k>0 and a(2)=-2, a(2k)=0 for all k>1. %D A131386 A. D. Aczel, Fermat's Last Theorem, Four Walls Eight Windows NY 1996 %D A131386 A. C. Clarke, The Last Theorem, Gollancz SF 2004. %D A131386 B. Cipra, What's Happening in the Mathematical Sciences 1994 Vol. 2, "A Truly Remarkable Proof" pp. 3-8 AMS Providence RI. %H A131386 Article number 1 %H A131386 Article number 2http://www.ansijournals.com/aja/2008/15-24.pdf %H A131386 A short form proof of FLT %F A131386 a(1)=1, a(2)=-2, a(2k+1)=1, a(2k)=0, k\geq{1}. %e A131386 We calculate x_2, y_2, z_2, u_2, and x_3, y_3, z_3, u_3, we discover (tere is no formula) a(1)=1, a(2)=-2, and x_4, y_4, z_4, u_4, we discover a(3), a(4), etc... %K A131386 easy,nonn,new %O A131386 1,2 %A A131386 Jamel Ghanouchi (jamel.ghanouchi(AT)topnet.tn), Aug 26 2008 %I A131180 %S A131180 1,2,4,14,53,313,2170,31896,387802,11507269,290487893,15905409672, %T A131180 816544187208,90196956861272,9093799327168995,2023727669041604444 %N A131180 Number of ways of placing non-attacking knights on a n X n chessboard symmetric under 180 degree rotation. %K A131180 nonn,new %O A131180 0,2 %A A131180 Ron Hardin (rhh(AT)cadence.com), Aug 25 2008 %I A130728 %S A130728 1,2,2,6,3,27,28,324,338,5991,6875,228512,322056,17782554,30487207, %T A130728 2704389848,5451449965,819155081866,1981437279113,497332226754331, %U A130728 1457925225531230,603767469974886673,2133983730438871395 %N A130728 Number of ways of placing non-attacking knights on a n X n chessboard symmetric under 90 degree rotation. %K A130728 nonn,new %O A130728 0,2 %A A130728 Ron Hardin (rhh(AT)cadence.com), Aug 25 2008 %I A130712 %S A130712 1,2,2,6,3,21,12,118,56,959,333,9892,3038,154506,38869,3291314,689349, %T A130712 103050492,17202265,4328976863,615532072,271539093065,30694296121, %U A130712 22908330863086,2181826599454,2875160955130336,217661432837240 %N A130712 Number of ways of placing non-attacking knights on a n X n chessboard symmetric about the diagonal and under 90 degree rotation. %K A130712 nonn,new %O A130712 0,2 %A A130712 Ron Hardin (rhh(AT)cadence.com), Aug 25 2008 %I A130666 %S A130666 1,2,4,10,21,89,248,1652,5760,59789,268529,4367552,24922412,641484896, %T A130666 4607113999,188770443286,1702258930069,111575238563044,1256539208451845, %U A130666 132483845142611245,1854392449274745630,316017552098202250347 %N A130666 Number of ways of placing non-attacking knights on a n X n chessboard symmetric about both diagonal and antidiagonal. %K A130666 nonn,new %O A130666 0,2 %A A130666 Ron Hardin (rhh(AT)cadence.com), Aug 25 2008 %I A130613 %S A130613 1,2,8,26,125,1089,11632,180936,3829276,122898169,5157005541, %T A130613 328952820944,27732644636492,3546034350173724,597955340246293229, %U A130613 153413017541604438052,51883480594556703445379 %N A130613 Number of ways of placing non-attacking knights on a n X n chessboard symmetric about main diagonal. %K A130613 nonn,new %O A130613 0,2 %A A130613 Ron Hardin (rhh(AT)cadence.com), Aug 25 2008 %I A129898 %S A129898 1,2,2,10,7,63,38,1252,456,42951,11051,2411116,461788,300699154, %T A129898 38535499,76231743090,6586274655,36545866507976,2160036716063, %U A129898 34857588279065543,1397443585027894,67527452467684091249 %N A129898 Number of ways of placing non-attacking knights on a n X n chessboard symmetric under horizontal and vertical reflection. %K A129898 nonn,new %O A129898 0,2 %A A129898 Ron Hardin (rhh(AT)cadence.com), Aug 25 2008 %I A129894 %S A129894 1,2,4,26,35,821,1462,148176,221880,96574099,126381799,196582061878, %T A129894 269726664738,1805524692669120,2273457768078921,66873306557904082902, %U A129894 76981272132177635155,9380430300181898788236006 %N A129894 Number of ways of placing non-attacking knights on a n X n chessboard symmetric under horizontal reflection. %K A129894 nonn,new %O A129894 0,2 %A A129894 Ron Hardin (rhh(AT)cadence.com), Aug 25 2008 %I A129858 %S A129858 6,12,12,20,21,20,30,32,32,30,42,45,45,45,42,56,60,59,59,60,56,72,77,74, %T A129858 72,74,77,72,90,96,90,84,84,90,96,90,110,117,107,95,90,95,107,117,110, %U A129858 132,140,125,105,92,92,105,125,140,132,156,165,144,114,90,81,90,114,144 %N A129858 A triangle of coefficients based on A000217: a(n)=Binomial[n+2,2]; t(n,m)=a(n - m + 1)*a(m + 1) - a((n - m + 1)*(m + 1)). %C A129858 Row sums are: %C A129858 {6, 24, 61, 124, 219, 350, 518, 720, 948, 1188, 1419}. %D A129858 G. E. Andrews, Number Theory, 1971, Dover Publications New York, p 44,p 85. %F A129858 a(n)=Binomial[n+2,2]; t(n,m)=a(n - m + 1)*a(m + 1) - a((n - m + 1)*(m + 1)). %e A129858 {6}, %e A129858 {12, 12}, %e A129858 {20, 21, 20}, %e A129858 {30, 32, 32, 30}, %e A129858 {42, 45, 45, 45, 42}, %e A129858 {56, 60, 59, 59, 60, 56}, %e A129858 {72, 77, 74, 72, 74, 77, 72}, %e A129858 {90, 96, 90, 84, 84, 90, 96, 90}, %e A129858 {110, 117, 107, 95, 90, 95, 107, 117, 110}, %e A129858 {132, 140, 125, 105, 92, 92, 105, 125, 140, 132}, %e A129858 {156, 165, 144, 114, 90, 81, 90, 114, 144, 165, 156} %t A129858 Clear[a, n, m, t] (*A000217*) a[0] = 1; a[1] = 3; a[n_] := a[n] = Binomial[n + 2, 2]; Table[a[n], {n, 0, 30}]; t[n_, m_] = FullSimplify[a[n - m + 1]*a[m + 1] - a[(n - m + 1)*(m + 1)]]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%] %Y A129858 Cf. A000217. %K A129858 nonn,tabl,new %O A129858 1,1 %A A129858 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 25 2008 %I A129855 %S A129855 9,18,18,30,36,30,45,60,60,45,63,90,100,90,63,84,126,150,150,126,84,108, %T A129855 168,210,225,210,168,108,135,216,280,315,315,280,216,135,165,270,360, %U A129855 420,441,420,360,270,165,198,330,450,540,588,588,540,450,330,198,234 %N A129855 A symmetrical triangle of coefficients based on A000217: a(n) = Binomial[n + 2, 2]; t(n,m)=a(n - m + 1)*a(m + 1). %C A129855 Row sums are: %C A129855 {9, 36, 96, 210, 406, 720, 1197, 1892, 2871, 4212, 6006}. %D A129855 G. E. Andrews, Number Theory, 1971, Dover Publications New York, p 44. %F A129855 a(n) = Binomial[n + 2, 2]; t(n,m)=a(n - m + 1)*a(m + 1). %e A129855 {9}, %e A129855 {18, 18}, %e A129855 {30, 36, 30}, %e A129855 {45, 60, 60, 45}, %e A129855 {63, 90, 100, 90, 63}, %e A129855 {84, 126, 150, 150, 126, 84}, %e A129855 {108, 168, 210, 225, 210, 168, 108}, %e A129855 {135, 216, 280, 315, 315, 280, 216, 135}, %e A129855 {165, 270, 360, 420, 441, 420, 360, 270, 165}, %e A129855 {198, 330, 450, 540, 588, 588, 540, 450, 330, 198}, %e A129855 {234, 396, 550, 675, 756, 784, 756, 675, 550, 396, 234} %t A129855 Clear[a, n, m, t] (*A000217*) a[0] = 1; a[1] = 3; a[n_] := a[n] = Binomial[n + 2, 2]; Table[a[n], {n, 0, 30}]; t[n_, m_] = a[n - m + 1]*a[m + 1]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%] %Y A129855 Cf. A000217. %K A129855 nonn,tabl,new %O A129855 1,1 %A A129855 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 25 2008 %I A129742 %S A129742 0,0,2,51,164945,18423138,615376173176,168483518571789, %T A129742 24434798429947993043,5256695596753687250025931034, %U A129742 4278271932454694494134007741935 %N A129742 Numbers of the form: a(n)=((Prime[n] - 1)! - (Prime[n] - 1))/(2*Prime[n]). %C A129742 From the proof of Sir John Wilson's theorem: %C A129742 numbers of sets of stellated p-gons. %D A129742 G. E. Andrews, Number Theory, 1971, Dover Publications New York, p 39. %F A129742 a(n)=((Prime[n] - 1)! - (Prime[n] - 1))/(2*Prime[n]). %t A129742 f[n_] = ((Prime[n] - 1)! - (Prime[n] - 1))/(2*Prime[n]); Table[f[n], {n, 1, 20}] %K A129742 nonn,new %O A129742 1,3 %A A129742 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 25 2008 %I A129413 %S A129413 1,6,9,14,18,25,38,51,71,89,116 %N A129413 The smallest value of a magic sum amongst all edge-magic injections of the complete graph K_n on n vertices. %D A129413 J. P. McSorley and J. A. Trono, On k-minimum and m-minimum Edge-Magic Injections of Graphs. Preprint, (2008). %D A129413 W. D. Wallis. Magic Graphs. Birkhauser, (2001). Section 2.3.3. %D A129413 W. D. Wallis, E. T. Baskoro, M. Miller and Slamin. Edge-Magic Total Labellings. Australas. J. Comb. v.22, (2000), pp.177-190. Section 7.1. %e A129413 a(3)=9 because in an edge-magic injection of the complete graph K_3 %e A129413 the smallest that the largest label used can be is 6. %e A129413 Then the other two labels sum to at least 1+2. %e A129413 Hence the smallest that the magic sum can be is 6+1+2=9, %e A129413 and such an edge-magic injection of K_3 with magic sum 9 exists. %K A129413 nonn,new %O A129413 1,2 %A A129413 John P. McSorley (mcsorley60(AT)hotmail.com), Aug 25 2008 %I A129367 %S A129367 36,120,120,300,400,300,630,1000,1000,630,1176,2100,2500,2100,1176,2016, %T A129367 3920,5250,5250,3920,2016,3240,6720,9800,11025,9800,6720,3240,4950, %U A129367 10800,16800,20580,20580,16800,10800,4950,7260,16500,27000,35280,38416 %N A129367 A symmetrical triangle of coefficient weights: A002415 :f(n)=n^2*(n^2 - 1)/12; t(n,m)=f(n - m + 1)*f(m + 1). %C A129367 Row sums with zeros: %C A129367 {0, 0, 36, 240, 1000, 3260, 9052, 22372, 50545}. %D A129367 Steven Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley and Sons, Inc. , New York, 1972, page145: Number of components from curvature R(i,j,k,l) :A002415. %F A129367 f(n)=n^2*(n^2 - 1)/12; t(n,m)=f(n - m + 1)*f(m + 1). %e A129367 Initial Zeros removed: %e A129367 {36}, %e A129367 {120, 120}, %e A129367 {300, 400, 300}, %e A129367 {630, 1000, 1000, 630}, %e A129367 {1176, 2100, 2500, 2100, 1176}, %e A129367 {2016, 3920, 5250, 5250, 3920, 2016}, %e A129367 {3240, 6720, 9800, 11025, 9800, 6720, 3240}, %e A129367 {4950, 10800, 16800, 20580, 20580, 16800, 10800, 4950}, %e A129367 {7260, 16500, 27000, 35280, 38416, 35280, 27000, 16500, 7260} %t A129367 f[n_] = n*(n - 1)*(n - 2)*(n + 3)/12; t[n_, m_] = f[n - m + 1]*f[m + 1]; Table[Table[t[n, m], {m, 2, n - 2}], {n, 2, 12}]; Flatten[%] %Y A129367 Cf.A117662, A002415. %K A129367 nonn,tabl,new %O A129367 1,1 %A A129367 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 25 2008 %I A118796 %S A118796 2,13,23,29,37,41,79,107,127,149,211,239,263,313,383,389,397,439,467, %T A118796 509,547,631,673,743,757,773,787,827,829,907,997,1019,1061,1091,1231, %U A118796 1297,1367,1451,1459,1487,1543,1559,1601,1609,1613,1627,1637,1699,1721 %N A118796 Prime numbers as : Prime * Glaisher = Round[Prime]. %t A118796 lst={};Do[p=Prime[n];p1=p*Glaisher;p2=Round[p1];If[Abs[p2-p1]<1&&PrimeQ[p2],AppendTo[lst,p]],{n,10^3}];lst %K A118796 nonn,uned,new %O A118796 1,1 %A A118796 Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 25 2008 %I A118786 %S A118786 2,5,7,31,71,83,89,101,103,109,139,223,241,293,349,433,491,509,521,541, %T A118786 599,617,641,719,751,787,827,883,947,997,1213,1291,1303,1321,1367,1381, %U A118786 1423,1531,1571,1597,1747,1787,2017,2027,2029,2111,2129,2207,2237,2341 %N A118786 Prime numbers as : Prime * Khinchin = Round[Prime]. %t A118786 lst={};Do[p=Prime[n];p1=p*Khinchin;p2=Round[p1];If[Abs[p2-p1]<1&&PrimeQ[p2],AppendTo[lst,p]],{n,10^3}];lst %K A118786 nonn,uned,new %O A118786 1,1 %A A118786 Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 25 2008 %I A118625 %S A118625 2,3,5,19,47,67,73,97,139,163,211,263,281,307,379,401,433,457,479,499, %T A118625 503,523,569,571,641,647,719,739,811,859,883,1187,1193,1259,1289,1409, %U A118625 1423,1427,1499,1571,1619,1637,1663,1787,1879,1901,1907,1951,1999,2089 %N A118625 Prime numbers as : Prime * Catalan = Round[Prime]. %t A118625 lst={};Do[p=Prime[n];p1=p*Catalan;p2=Round[p1];If[Abs[p2-p1]<1&&PrimeQ[p2],AppendTo[lst,p]],{n,10^3}];lst %K A118625 nonn,uned,new %O A118625 1,1 %A A118625 Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 25 2008 %I A118546 %S A118546 9,42,42,120,196,120,270,560,560,270,525,1260,1600,1260,525,924,2450, %T A118546 3600,3600,2450,924,1512,4312,7000,8100,7000,4312,1512,2340,7056,12320, %U A118546 15750,15750,12320,7056,2340,3465,10920,20160,27720,30625,27720,20160 %N A118546 A symmetrical triangle of coefficient weights: A117662 :f(n)=n*(n - 1)*(n - 2)*(n + 3)/12; t(n,m)=f(n - m + 1)*f(m + 1). %C A118546 Row sums with zeros: %C A118546 {0, 0, 9, 84, 436, 1660, 5170, 13948, 33748}. %D A118546 Steven Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley and Sons, Inc. , New York, 1972, page145: Number of algebraic scalars constructed from curvature R(i,j,k,l) and metric ground form g(i,j):A117662. %F A118546 f(n)=n*(n - 1)*(n - 2)*(n + 3)/12; t(n,m)=f(n - m + 1)*f(m + 1). %e A118546 Initial Zeros removed: %e A118546 {9}, %e A118546 {42, 42}, %e A118546 {120, 196, 120}, %e A118546 {270, 560, 560, 270}, %e A118546 {525, 1260, 1600, 1260, 525}, %e A118546 {924, 2450, 3600, 3600, 2450, 924}, %e A118546 {1512, 4312, 7000, 8100, 7000, 4312, 1512}, %e A118546 {2340, 7056, 12320, 15750, 15750, 12320, 7056, 2340}, %e A118546 {3465, 10920, 20160, 27720, 30625, 27720, 20160, 10920, 3465} %t A118546 f[n_] = n*(n - 1)*(n - 2)*(n + 3)/12; t[n_, m_] = f[n - m + 1]*f[m + 1]; Table[Table[t[n, m], {m, 2, n - 2}], {n, 2, 12}]; Flatten[%] %Y A118546 Cf.A117662. %K A118546 nonn,tabl,new %O A118546 1,1 %A A118546 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 25 2008 %I A118484 %S A118484 3,5,19,23,29,53,71,103,127,137,179,227,241,283,313,331,397,487,491,601, %T A118484 647,709,751,761,809,829,881,937,947,1051,1069,1093,1229,1259,1301,1319, %U A118484 1381,1423,1433,1453,1489,1571,1579,1609,1693,1787,1789,1901,1997,2029 %N A118484 Prime numbers as : Prime * EulerGamma = Round[Prime]. %t A118484 lst={};Do[p=Prime[n];p1=p*EulerGamma;p2=Round[p1];If[Abs[p2-p1]<1&&PrimeQ[p2],AppendTo[lst,p]],{n,10^3}];lst %K A118484 nonn,uned,new %O A118484 1,1 %A A118484 Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 25 2008 %I A117763 %S A117763 2,3,7,19,23,29,97,101,103,107,149,181,227,311,353,379,433,457,563,631, %T A117763 719,761,883,919,941,971,997,1049,1087,1223,1279,1291,1297,1321,1427, %U A117763 1447,1453,1531,1627,1699,1831,1861,1877,2039,2143,2207,2213,2239,2269 %N A117763 Prime numbers as : Prime * GoldenRatio = Round[Prime]. %t A117763 lst={};Do[p=Prime[n];p1=p*GoldenRatio;p2=Round[p1];If[Abs[p2-p1]<1&&PrimeQ[p2],AppendTo[lst,p]],{n,10^3}];lst %K A117763 nonn,uned,new %O A117763 1,1 %A A117763 Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 25 2008 %I A116968 %S A116968 2,7,29,37,71,113,163,179,199,227,283,439,463,503,541,547,647,733,761, %T A116968 823,839,887,953,1031,1049,1051,1093,1327,1399,1549,1627,1741,1847,1861, %U A116968 1901,1951,2017,2053,2081,2179,2221,2287,2309,2399,2477,2591,2689,2711 %N A116968 Prime numbers as : Prime * E = Round[Prime]. %t A116968 lst={};Do[p=Prime[n];p1=p*E;p2=Round[p1];If[Abs[p2-p1]<1&&PrimeQ[p2],AppendTo[lst,p]],{n,10^3}];lst %K A116968 nonn,uned,new %O A116968 1,1 %A A116968 Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 25 2008 %I A116671 %S A116671 13,17,31,53,71,73,101,181,197,223,229,239,241,281,311,313,353,367,491, %T A116671 521,607,821,859,863,919,1129,1217,1303,1381,1427,1471,1583,1667,1721, %U A116671 1723,1753,1811,1877,1879,1933,1979,2017,2063,2089,2221,2399,2441,2447 %N A116671 Prime numbers as : Prime * Pi = Round[Prime]. %e A116671 Pi*13~Round[41]; %e A116671 Pi*17~Round[53]; %t A116671 lst={}; Do[p=Prime[n]; p1=p*Pi; p2=Round[p1]; If[Abs[p2-p1]<1&&PrimeQ[p2],AppendTo[lst,p]],{n,10^3}]; lst %K A116671 nonn,uned,new %O A116671 1,1 %A A116671 Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 25 2008 %I A115400 %S A115400 6,96,780,4080,15330,45696,115416,257760,523710,987360,1752036,2957136, %T A115400 4785690,7472640,11313840,16675776,24006006,33844320 %N A115400 Number of n-colorings of the octahedral graph. %C A115400 The octahedral graph is the dual of the cubical graph whose chromatic polynomial is evaluated in A140986. %H A115400 Eric W. Weisstein, Octahedral Graph. %F A115400 a(n) = n*(n-1)*(n-2)*(n^3 - 9*n^2 + 29*n - 32). %Y A115400 Cf. A140986. %K A115400 easy,more,nonn,new %O A115400 3,1 %A A115400 Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 25 2008 %I A114896 %S A114896 1,2,2,2,4,2,3,4,4,3,2,6,4,6,2,4,4,6,6,4,4,2,8,4,9,4,8,2,4,4,8,6,6,8,4, %T A114896 4,3,8,4,12,4,12,4,8,3,4,6,8,6,8,8,6,8,6,4,2,8,6,12,4,16,4,12,6,8,2 %N A114896 A symmetrical triangle of weight coefficients using the Divisors Sigma function: t(n,m)=DivisorSigma[0, n - m + 1]*DivisorSigma[0, m + 1]. %C A114896 Row sums are: %C A114896 {1, 4, 8, 14, 20, 28, 37, 44, 58, 64, 80}. %F A114896 t(n,m)=DivisorSigma[0, n - m + 1]*DivisorSigma[0, m + 1]. %e A114896 {1}, %e A114896 {2, 2}, %e A114896 {2, 4, 2}, %e A114896 {3, 4, 4, 3}, %e A114896 {2, 6, 4, 6, 2}, %e A114896 {4, 4, 6, 6, 4, 4}, %e A114896 {2, 8, 4, 9, 4, 8, 2}, %e A114896 {4, 4, 8, 6, 6, 8, 4, 4}, %e A114896 {3, 8, 4, 12, 4, 12, 4, 8, 3}, %e A114896 {4, 6, 8, 6, 8, 8, 6, 8, 6, 4}, %e A114896 {2, 8, 6, 12, 4, 16, 4, 12, 6, 8, 2} %t A114896 t[n_, m_] =DivisorSigma[0, n - m + 1]*DivisorSigma[0, m + 1]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%] %Y A114896 Cf. A000005. %K A114896 nonn,tabl,new %O A114896 1,2 %A A114896 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 25 2008 %I A113793 %S A113793 1,1,1,2,1,2,2,2,2,2,4,2,4,2,4,2,4,4,4,4,2,6,2,8,4,8,2,6,4,6,4,8,8,4,6, %T A113793 4,6,4,12,4,16,4,12,4,6,4,6,8,12,8,8,12,8,6,4,10,4,12,8,24,4,24,8,12,4, %U A113793 10 %N A113793 A symmetrical triangle of weight coefficients using the Euler totient function: t(n,m)=EulerPhi[n - m + 1]*EulerPhi[m + 1]. %C A113793 Row sums are: %C A113793 {1, 2, 5, 8, 16, 20, 36, 44, 68, 76, 120} %F A113793 t(n,m)=EulerPhi[n - m + 1]*EulerPhi[m + 1]. %e A113793 {1}, %e A113793 {1, 1}, %e A113793 {2, 1, 2}, %e A113793 {2, 2, 2, 2}, %e A113793 {4, 2, 4, 2, 4}, %e A113793 {2, 4, 4, 4, 4, 2}, %e A113793 {6, 2, 8, 4, 8, 2, 6}, %e A113793 {4, 6, 4, 8, 8, 4, 6, 4}, %e A113793 {6, 4, 12, 4, 16, 4, 12, 4, 6}, %e A113793 {4, 6, 8, 12, 8, 8, 12, 8, 6, 4}, %e A113793 {10, 4, 12, 8, 24, 4, 24, 8, 12, 4, 10} %t A113793 t[n_, m_] = EulerPhi[n - m + 1]*EulerPhi[m + 1]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%] %Y A113793 Cf. A000010. %K A113793 nonn,tabl,new %O A113793 1,4 %A A113793 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 25 2008 %I A113607 %S A113607 1,2,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,11,11,1,1,1,1,26,66,26,1,1,1,1,57, %T A113607 302,302,57,1,1,1,1,120,1191,2416,1191,120,1,1,1,1,247,4293,15619,15619, %U A113607 4293,247,1,1,1,1,502,14608,88234,156190,88234,14608,502,1,1,1,1,1013 %N A113607 An extended triangle of Eulerian coefficients A123125: f(x,n)=x^(n+1)+1+A123125(x,n). %C A113607 Not entirely symmetrical, the x^(n+1)+1 polynomials was added to remove zeros and make the triangle more symmetrical. %C A113607 Row sums are: %C A113607 {1, 3, 3, 4, 8, 26, 122, 722, 5042, 40322, 362882, 3628802}. %F A113607 f(x,n)=x^(n+1)+1+A123125(x,n). %t A113607 lear[f, x, n, a] f[x_, n_] := f[x,n] = x^(n + 1) + (1 - x)^(n + 1)*Sum[k^n*x^k, {k, 0, Infinity}] + 1; Table[FullSimplify[ExpandAll[f[x, n]]], {n, 0, 10}]; a = Join[{{1}}, Table[CoefficientList[FullSimplify[ExpandAll[f[x, n]]], x], {n, 0, 10}]]; Flatten[a] %Y A113607 Cf. A123125. %K A113607 nonn,tabl,new %O A113607 1,2 %A A113607 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 25 2008 %I A113606 %S A113606 1,1,3,5,13,45,8237,35184372097069 %N A113606 Binary power sequence: a(n) = a(n-1) + 2^a(n-2). %C A113606 The ninth term is too large to show here. %Y A113606 Cf. A000225. %K A113606 nonn,new %O A113606 1,3 %A A113606 Matt Wynne (mattwyn(AT)verizon.net), Aug 25 2008 %I A113582 %S A113582 1,1,1,1,2,1,1,4,4,1,1,7,10,7,1,1,11,19,19,11,1,1,16,31,37,31,16,1,1,22, %T A113582 46,61,61,46,22,1,1,29,64,91,101,91,64,29,1,1,37,85,127,151,151,127,85, %U A113582 37,1,1,46,109,169,211,226,211,169,109,46,1 %N A113582 A symmetrical triangle of coefficients: t(n,m)=(n - m)*(n - m + 1)*m*(m + 1)/4 + 1. %C A113582 Row sums are: %C A113582 {1, 2, 4, 10, 26, 62, 133, 260, 471, 802, 1298}. %F A113582 t(n,m)=(n - m)*(n - m + 1)*m*(m + 1)/4 + 1. %e A113582 {1}, %e A113582 {1, 1}, %e A113582 {1, 2, 1}, %e A113582 {1, 4, 4, 1}, %e A113582 {1, 7, 10, 7, 1}, %e A113582 {1, 11, 19, 19, 11, 1}, %e A113582 {1, 16, 31, 37, 31, 16, 1}, %e A113582 {1, 22, 46, 61, 61, 46, 22, 1}, %e A113582 {1, 29, 64, 91, 101, 91, 64, 29, 1}, %e A113582 {1, 37, 85, 127, 151, 151, 127, 85, 37, 1}, %e A113582 {1, 46, 109, 169, 211, 226, 211, 169, 109, 46, 1} %t A113582 Clear[t, n, m] t[n_, m_] = (n - m)*(n - m + 1)*m*(m + 1)/4 + 1; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%] %K A113582 nonn,tabl,new %O A113582 1,5 %A A113582 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Aug 25 2008 %I A113280 %S A113280 1,1,1,1,10,1,1,25,25,1,1,46,65,46,1,1,73,121,121,73,1,1,106,193,226, %T A113280 193,106,1,1,145,281,361,361,281,145,1,1,190,385,526,577,526,385,190,1, %U A113280 1,241,505,721,841,841,721,505,241,1,1,298,641,946,1153,1226,1153,946 %N A113280 A symmetrical triangle of coefficients: t(n,m)=(n - m)*(n - m + 2)*m*(m + 2) + 1. %C A113280 Row sums are: %C A113280 {1, 2, 12, 52, 159, 390, 826, 1576, 2781, 4618, 7304}. %F A113280 t(n,m)=(n - m)*(n - m + 2)*m*(m + 2) + 1. %e A113280 {1}, %e A113280 {1, 1}, %e A113280 {1, 10, 1}, %e A113280 {1, 25, 25, 1}, %e A113280 {1, 46, 65, 46, 1}, %e A113280 {1, 73, 121, 121, 73, 1}, %e A113280 {1, 106, 193, 226, 193, 106, 1}, %e A113280 {1, 145, 281, 361, 361, 281, 145, 1}, %e A113280 {1, 190, 385, 526, 577, 526, 385, 190, 1}, %e A113280 {1, 241, 505, 721, 841, 841, 721, 505, 241, 1}, %e A113280 {1, 298, 641, 946, 1153, 1226, 1153, 946, 641, 298, 1} %t A113280 Clear[t, n, m] t[n_, m_] = (n - m)*(n - m + 2)*m*(m + 2) + 1; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%] %K A113280 nonn,tabl,new %O A113280 1,5 %A A113280 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Aug 25 2008 %I A113171 %S A113171 660,1092,1140,1155,1260,1320,1365,1380,1428,1540,1560,1740,1785,1820, %T A113171 1860 %N A113171 Short legs 'A' of exactly 7 primitive Pythagorean triangles. %F A113171 a^2+b^2=c^2 %e A113171 Examples of triples: 660.779.1021, 660.989.1189, 660.2989.3061, 660.4331.4381, 660.12091.12109, 660.27221.27229, 660.108899.108901 %e A113171 1092.1325.1717, 1092.1595.1933, 1092.6035.6133, 1092.8245.8317, 1092.33115.33133, 1092.74525.74533, 1092.298115.298117 %t A113171 PyphagoreanAs[a_]:=(q={};k=0;Do[y=(a^2+b^2)^0.5;c=IntegerPart[y];If[c==y,p=0;If[GCD[a,b,c]==1,AppendTo[q,a.b.c];k++ ]],{b,a+1,a^2}];PrependTo[q,k];q)lst={};Do[If[PyphagoreanAs[n][[1]]==7,Print[n];AppendTo[lst,n]],{n,6*10^2,2*10^3}];lst %Y A113171 Cf. A056866 Orders of non-solvable groups.. A093006 Referring to the triangle in A093005, sequence contains the least term with maximal number of divisors. A138605 Short legs of more than 3 primitive Pythagorean triangles. A033993 Numbers that are divisible by exactly four different primes. %K A113171 nonn,new %O A113171 1,1 %A A113171 Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 25 2008 %I A112927 %S A112927 1,3,7,5,31,1,127,17,73,11,23,13,8191,43,151,257,131071,19,524287,41, %T A112927 337,683,47,241,601,2731,262657,29,233,331,2147483647,65537,599479, %U A112927 43691,71,37,223,174763,79,61681 %N A112927 a(n) is the least prime such that the multiplicative order of 2 mod a(n) equals to n, or a(n)=1 if no such prime exists. %C A112927 If a(n) differs from 1, then a(n) is the minimal prime divisor of A064078(n); %C A112927 a(n)=n+1 iff n+1 is prime from A001122; a(n)=2n+1 iff 2n+1 is prime from A115591. %Y A112927 Cf. A002326 A064078 A001122 A115591. %K A112927 nonn,new %O A112927 1,2 %A A112927 Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 25 2008 %I A112857 %S A112857 1,1,1,1,3,1,1,7,5,1,1,15,17,7,1,1,31,49,31,9,1,1,63,129,111,49,11,1,1, %T A112857 127,321,351,209,71,13,1 %N A112857 Triangle read by rows: number of Green’s R-classes in the semigroup of order-preserving partial transformations (of an n-element chain) consisting of elements of height k (height(alpha) = |Im(alpha)|). . %C A112857 Sum of rows of R(n, k) is A007051, R(n,k) = |A118801| %D A112857 Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations. Journal of Algebra 278, (2004), 342-359. %D A112857 Laradji, A. and Umar, A. Combinatorial results for semigroups of order-decreasing partial transformations. J. Integer Seq. 7 (2004), 04.3.8, 14 pp %F A112857 R(n,k)=sum(j=p,n,C(n,j*C(j-1,p-1)) R(n,k)=2*R(n-1,k)+R(n-1,k-1), R(n,0)= 1 = R(n,n) %e A112857 R(3,2) = 5 because in a regular semigroup of transformations the Green’s R-classes coincide with convex partitions of subsets of {1,2,3} with convex classes (modulo the subsets): {1}, {2}/{1}, {3}/{2}, {3}/{1,2}, {3}/{1}, {2,3} %Y A112857 Cf. A007051, A118801. %K A112857 nonn,tabl,new %O A112857 0,5 %A A112857 A. Umar (aumarh(AT)squ.edu.om), Aug 25 2008 %I A112091 %S A112091 1,2,6,21,76,276,1001,3626,13126,47501,171876,621876,2250001,8140626, %T A112091 29453126,106562501,385546876,1394921876,5046875001,18259765626, %U A112091 66064453126 %N A112091 Number of idempotent order-preserving partial transformations (of an n-element chain). %D A112091 Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations. Journal of Algebra 278, (2004), 342-359. %F A112091 a(n)= ((sqrt(5))^(n-1))*(((sqrt(5)+1)/2)^n-((sqrt(5)-1)/2)^n)); a(n)=1+5*(a(n-1)-a(n-2)), a(0)=1, a(1)=2 %e A112091 a(2) = 6 because there are exactly 6 idempotent order-preserving partial transformations (on a 2-element chain), namely: the empty map, (1)->(1), (2)->(2), (1,2)->(1,1), (1,2)->(1,2), (1,2)->(2,2)– the mappings are coordinate-wise %K A112091 nonn,new %O A112091 0,2 %A A112091 A. Umar (aumarh(AT)squ.edu.om), Aug 25 2008 %I A111776 %S A111776 1,1,1,1,2,3,1,4,6,10,1,8,12,20,35,1,16,24,40,70,125,1,32,48,80,140,250, %T A111776 450,1,64,96,160,280,500,900,1625 %N A111776 Triangle read by rows: number of idempotent order-preserving partial transformations (of an n-element chain) of waist k (waist(alpha) = max(Im(alpha)). %C A111776 G(n, n) is A081567(n – 1) %D A111776 Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations. Journal of Algebra 278, (2004), 342-359. %F A111776 G(n,k)= (2^(n-k))*G(n,n)=(2^(n-k))*A081567(n-1), G(0,0) = 1 %e A111776 G(3,2) = 6 because there are exactly 6 idempotent order-preserving partial transformations (on a 3-element chain) of waist 2, namely: (2)->(2), (1,2)->(1,2), (1,2)->(2,2),(1,3)->(3,3), (2,3)->(2,2), (2,3)->(3,3) – the mappings are coordinate-wise %Y A111776 Cf. A081567. %K A111776 nonn,tabl,new %O A111776 0,5 %A A111776 A. Umar (aumarh(AT)squ.edu.om), Aug 25 2008 %I A111589 %S A111589 1,1,1,1,2,3,1,3,9,8,1,4,18,32,21,1,5,30,80,105,55,1,6,45,160,315,330, %T A111589 144,1,7,63,280,735,1155,1008,377 %N A111589 Triangle read by rows: number of idempotent order-preserving partial transformations (of an n-element totally ordered set) of height k (height(alpha) = |Im(alpha)|). %C A111589 F(n; n) is A001906(n – 1) %D A111589 Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations. Journal of Algebra 278, (2004), 342-359 %F A111589 F(n,k)= C(n,k)*A001906(k-1), (n>=k>0),F(0,0)=1 %e A111589 F(3,2) = 9 because there are exactly 9 idempotent order-preserving partial transformations (on a 3-element chain) of width 2, namely: (1,2)->(1,1), (1,2)->(1,2), (1,2)->(2,2), (1,2)->(3,3), (1,3)->(1,1), (1,3)->(1,3),(1,3)->(3,3), (2,3)->(2,2), (2,3)->(2,3),( 2,3)->(3,3) – the mappings are coordinate-wise %Y A111589 Cf. A001906(n – 1). %K A111589 nonn,tabl,new %O A111589 0,5 %A A111589 A. Umar (aumarh(AT)squ.edu.om), Aug 25 2008 %I A111516 %S A111516 1,1,1,1,3,4,1,7,12,18,1,15,32,56,88,1,31,80,160,280,450,1,63,192,432, %T A111516 832,1452,2364,1,127,448,1120,2352,4244,7700,12642 %N A111516 Triangle read by rows: number of order-preserving partial transformations (of an n-element totally ordered set) of waist k (waist(alpha) = max(Im(alpha)). %C A111516 G(n; n) is A050146 and sum(k=1,n,G(n; k)) is A123164 %D A111516 Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations. Journal of Algebra 278, (2004), 342-359. %F A111516 G(n,k)=sum(j=0,k,C(n,j)*C(k+j-2,j-1)); G(n,k)=2*G(n-1,k)-G(n-1,k-1)+G(n,k-1), G(n,0)=1 (n>=0), G(0,k)=0 (k>0) %e A111516 G(2,2) = 4 because there are exactly 4 order-preserving partial transformations (on a 2-element chain) of waist 2, namely: (1)->(2), (2)->(2),(1,2)->(1,2),(1,2)->(2,2) – the mappings are coordinate-wise %Y A111516 Cf. A050146, A123164. %K A111516 nonn,tabl,new %O A111516 0,5 %A A111516 A. Umar (aumarh(AT)squ.edu.om), Aug 25 2008 %I A111008 %S A111008 1,1,1,1,1,5,1,7,1,1,1,11,1,13,7,5,1,17,1 %N A111008 Bernoulli A000367/Debye A141590. %C A111008 1's with only primes? See A141517. %K A111008 nonn,uned,new %O A111008 0,6 %A A111008 Paul Curtz (bpcrtz(AT)free.fr), Aug 25 2008 %I A110858 %S A110858 1,1,1,1,2,2,1,3,6,6,1,4,12,24,20,1,5,20,60,100,70,1,6,30,120,300,420, %T A110858 252,1,7,42,210,700,1470,1764,924 %N A110858 Triangle read by rows: number of order-preserving partial transformations (of an n-element chain) of width and waist both equal to r (width(alpha) = |Dom(alpha)| and waist(alpha) = max(Im(alpha)). %D A110858 Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations. Journal of Algebra 278, (2004), 342-359 %F A110858 G(n,k)=C(n,k)*C(2*k-2,k-1), n >=k >0 %e A110858 G(3,2)=6 because there are exactly 6 order-preserving partial transformations (on a 3-element chain) of both width and waist equal to 2, namely: (1,2)->(1,2),(1,2)->(2,2),(1,3)->(1,2),(1,3)->(2,2),(2,3)->(1,2),(2,3)->(2,2) %K A110858 nonn,tabl,new %O A110858 1,5 %A A110858 A. Umar (aumarh(AT)squ.edu.om), Aug 25 2008 %I A110854 %S A110854 1,0,0,4,0,4,4,4,2,2,0,2,0,0,0,2,4,0,4,0,0,10,10,4,4,4,4,2,6,6,0,2,6,4, %T A110854 0,2,6,0,6,0,2,4,6,10,8,0,8,6,8,4 %V A110854 1,0,0,4,0,-4,4,-4,2,2,0,-2,0,0,0,-2,4,0,-4,0,0,10,-10,4,4,-4,-4,2,6,-6,0,2,6,4,0,-2,6, %W A110854 0,-6,0,2,4,-6,10,-8,0,8,6,-8,-4 %N A110854 First bisection of primes(A000040)= 4, 6, 6, 10, 8, 6, 10, 8, 10, 8 - second (3, 6, 6, 6, 8, 10, 6, 12, 8, 6). %C A110854 (Bisections submitted Aug 24 2008). Ordered different terms: A004275? %K A110854 sign,uned,new %O A110854 0,4 %A A110854 Paul Curtz (bpcrtz(AT)free.fr), Aug 25 2008 %I A110361 %S A110361 1,1,1,4,1,4,6,4,4,6,15,6,16,6,15,32,15,24,24,15,32,65,32,60,36,60,32, %T A110361 65,147,65,128,90,90,128,65,147,306,147,260,192,225,192,260,147,306,660, %U A110361 306,588,390,480,480,390,588,306,660,1424,660,1224,882,975,1024,975,882 %N A110361 A triangle of coefficients based on A000931 and A000045: a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)*a(m + 1)*Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)]. %C A110361 Row sums are: %C A110361 {1, 2, 9, 20, 58, 142, 350, 860, 2035, 4848, 11354}. %F A110361 a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)*a(m + 1)*Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)]. %e A110361 {1}, %e A110361 {1, 1}, %e A110361 {4, 1, 4}, %e A110361 {6, 4, 4, 6}, %e A110361 {15, 6, 16, 6, 15}, %e A110361 {32, 15, 24, 24, 15, 32}, %e A110361 {65, 32, 60, 36, 60, 32, 65}, %e A110361 {147, 65, 128, 90, 90, 128, 65, 147}, %e A110361 {306, 147, 260, 192, 225, 192, 260, 147, 306}, %e A110361 {660, 306, 588, 390, 480, 480, 390, 588, 306, 660}, %e A110361 {1424, 660, 1224, 882, 975, 1024, 975, 882, 1224, 660, 1424} %t A110361 Clear[t, a, n, m] a[0] = 1; a[1] = 1; a[2] = 1; a[n_] := a[n] = a[n - 2] + a[n - 3]; t[n_, m_] := a[(n - m + 1)]* a[(m + 1)]*Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%] %Y A110361 Cf. A141611, A141617, A000931, A000045, A058071. %K A110361 nonn,tabl,new %O A110361 1,4 %A A110361 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 24 2008 %I A110102 %S A110102 1,1,1,2,1,2,2,2,2,2,3,2,4,2,3,4,3,4,4,3,4,5,4,6,4,6,4,5,7,5,8,6,6,8,5, %T A110102 7,9,7,10,8,9,8,10,7,9,12,9,14,10,12,12,10,14,9,12,16,12,18,14,15,16,15, %U A110102 14,18,12,16 %N A110102 A triangle of coefficients based on A000931: a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)*a(m + 1). %C A110102 Row sums are: %C A110102 {1, 2, 5, 8, 14, 22, 34, 52, 77, 114, 166} %F A110102 a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)*a(m + 1). %e A110102 {1}, %e A110102 {1, 1}, %e A110102 {2, 1, 2}, %e A110102 {2, 2, 2, 2}, %e A110102 {3, 2, 4, 2, 3}, %e A110102 {4, 3, 4, 4, 3, 4}, %e A110102 {5, 4, 6, 4, 6, 4, 5}, %e A110102 {7, 5, 8, 6, 6, 8, 5, 7}, %e A110102 {9, 7, 10, 8, 9, 8, 10, 7, 9}, %e A110102 {12, 9, 14, 10, 12, 12, 10, 14, 9, 12}, %e A110102 {16, 12, 18, 14, 15, 16, 15, 14, 18, 12, 16} %t A110102 Clear[t, a, n, m] a[0] = 1; a[1] = 1; a[2] = 1; a[n_] := a[n] = a[n - 2] + a[n - 3]; t[n_, m_] := a[(n - m + 1)]*a[(m + 1)]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%] %Y A110102 Cf. A141611, A141617, A000931. %K A110102 nonn,tabl,new %O A110102 1,4 %A A110102 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 24 2008 %I A110023 %S A110023 1,1,1,2,2,2,2,6,6,2,3,8,24,8,3,4,15,40,40,15,4,5,24,90,80,90,24,5,7,35, %T A110023 168,210,210,168,35,7,9,56,280,448,630,448,280,56,9,12,81,504,840,1512, %U A110023 1512,840,504,81,12,16,120,810,1680,3150,4032,3150,1680,810,120,16 %N A110023 A triangle of coefficients based on A000931 and the Pascal's triangle: a(n)=a(n-2)+a(n-3); t(n,m)=a(n - m + 1)*a(m + 1)*Binomial[n, m]. %C A110023 Row sums are: %C A110023 {1, 2, 6, 16, 46, 118, 318, 840, 2216, 5898, 15584} %F A110023 a(n)=a(n-2)+a(n-3); t(n,m)=a(n - m + 1)*a(m + 1)*Binomial[n, m]. %e A110023 {1}, %e A110023 {1, 1}, %e A110023 {2, 2, 2}, %e A110023 {2, 6, 6, 2}, %e A110023 {3, 8, 24, 8, 3}, %e A110023 {4, 15, 40, 40, 15, 4}, %e A110023 {5, 24, 90, 80, 90, 24, 5}, %e A110023 {7, 35, 168, 210, 210, 168, 35, 7}, %e A110023 {9, 56, 280, 448, 630, 448, 280, 56, 9}, %e A110023 {12, 81, 504, 840, 1512, 1512, 840, 504, 81, 12}, %e A110023 {16, 120, 810, 1680, 3150, 4032, 3150, 1680, 810, 120, 16} %t A110023 Clear[t, a, n, m] a[0] = 1; a[1] = 1; a[2] = 1; a[n_] := a[n] = a[n - 2] + a[n - 3]; t[n_, m_] := a[(n - m + 1)]*a[(m + 1)]*Binomial[n, m]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}] Flatten[%] %Y A110023 Cf. A141611, A141617, A000931. %K A110023 nonn,tabl,new %O A110023 1,4 %A A110023 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 24 2008 %I A109906 %S A109906 1,1,1,2,2,2,3,6,6,3,5,12,24,12,5,8,25,60,60,25,8,13,48,150,180,150,48, %T A109906 13,21,91,336,525,525,336,91,21,34,168,728,1344,1750,1344,728,168,34,55, %U A109906 306,1512,3276,5040,5040,3276,1512,306,55,89,550,3060,7560,13650,16128 %N A109906 A triangle of coefficients based on A000045 and the Pascal's triangle: t(n,m)=Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)]*Binomial[n, m]. %C A109906 Row sums are: %C A109906 {1, 2, 6, 18, 58, 186, 602, 1946, 6298, 20378, 65946} %F A109906 t(n,m)=Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)]*Binomial[n, m]. %e A109906 {1}, %e A109906 {1, 1}, %e A109906 {2, 2, 2}, %e A109906 {3, 6, 6, 3}, %e A109906 {5, 12, 24, 12, 5}, %e A109906 {8, 25, 60, 60, 25, 8}, %e A109906 {13, 48, 150, 180, 150, 48, 13}, %e A109906 {21, 91, 336, 525, 525, 336, 91, 21}, %e A109906 {34, 168, 728, 1344, 1750, 1344, 728, 168, 34}, %e A109906 {55, 306, 1512, 3276, 5040, 5040, 3276, 1512, 306, 55}, %e A109906 {89, 550, 3060, 7560, 13650, 16128, 13650, 7560, 3060, 550, 89} %t A109906 Clear[t, n, m] t[n_, m_] := Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)]*Binomial[n, m]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%] %Y A109906 Cf. A141611, A141617, A000045. %K A109906 nonn,tabl,new %O A109906 1,4 %A A109906 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 24 2008 %I A143679 %S A143679 1,1,1,1,8,1,1,15,15,1,1,22,78,22,1,1,29,190,190,29,1,1,36,351,848,351, %T A143679 36,1,1,43,561,2339,2339,561,43,1,1,50,820,5006,9766,5006,820,50,1,1,57, %U A143679 1128,9192,28806 %N A143679 Pascal-(1,6,1) array. %F A143679 Number triangle T(n,k)=sum{j=0..n-k, binomial(n-k,j)binomial(k,j)7^j} Riordan array (1/(1-x),x(1+6x)/(1-x)). As a square array read by diagonals T(n, 0)=T(0, k)=1, T(n, k)=T(n, k-1)+6T(n-1, k-1)+T(n-1, k). %e A143679 Triangle begins %e A143679 1, %e A143679 1, 1, %e A143679 1, 8, 1, %e A143679 1, 15, 15, 1, %e A143679 1, 22, 78, 22, 1, %e A143679 1, 29, 190, 190, 29, 1, %e A143679 1, 36, 351, 848, 351, 36, 1, %e A143679 1, 43, 561, 2339, 2339, 561, 43, 1, %e A143679 ... %K A143679 easy,nonn,tabl,new %O A143679 0,5 %A A143679 Paul Barry (pbarry(AT)wit.ie), Aug 28 2008 %I A143674 %S A143674 1,1,2,4,17,379 %N A143674 Number of maximal antichains in the poset of Dyck paths ordered by inclusion. %C A143674 Maximal antichains are those which cannot be extended without violating the antichain condition. %C A143674 This is the breakdown by size of (or number of elements in) the antichains beginning with antichains of size 0 and increasing: %C A143674 n=0 1 %C A143674 n=1 0,1 %C A143674 n=2 0,2 %C A143674 n=3 0,3,1 %C A143674 n=4 0,3,8,6 %C A143674 n=5 0,3,14,62,132,124,42,2 %D A143674 R. P. Stanley, Enumerative Combinatorics 1, Cambridge University Press, New York, 1997. %H A143674 J. Woodcock, Properties of the poset of Dyck paths ordered by inclusion %e A143674 For n = 3 there are 4 maximal antichains. Assume that the five elements in the D_3 poset are depicted using a Hasse diagram, and labelled A through E from bottom to top. Then the 4 maximal antichains are: {A}, {B,C}, {D}, {E}. %Y A143674 Cf. A143672.