0,4

The result that number of partitions of n into distinct parts = number of partitions of n into odd parts is due to Euler.

Bijection: given n = l1 * 1 + l2 * 3 + l3 * 5 + l7 * 7 + ..., a partition into odd parts, write each li in binary, li = 2^a1 + 2^a2 + 2^a3 + ... where the aj's are all different, then expand n = (2^a1 * 1 + ...)*1 + ... by removing the brackets and we get a partition into distinct parts. For the reverse operation, just keep splitting any even number into halves until no evens remain.

Euler transform of period 2 sequence [1,0,1,0,...]. - Michael Somos, Dec 16, 2002

Number of different partial sums 1+[1,2]+[1,3]+[1,4]+..., where [1,x] indicates a choice. e.g. a(6)=4, as we can write 1+1+1+1+1+1, 1+2+3, 1+2+1+1+1, 1+1+3+1. - Jon Perry (perry(AT)globalnet.co.uk), Dec 31 2003

a(n) is the sum of the number of partitions of x_j into at most j parts, where j is the index for the j-th triangular number and n-T(j)=x_j. For example; a(12)=partitions into <=4 parts of 12-T(4)=2 + partitions into <=3 parts of 12-T(3)=6 + partitions into <=2 parts of 12-T(2)=9 + partitions into 1 part of 12-T(1)=11 =(2)(11)+(6)(51)(42)(411)(33)(321)(222)+(9)(81)(72)(63)(54)+(11) =2+7+5+1 =15 - Jon Perry (perry(AT)globalnet.co.uk), Jan 13 2004

Number of partitions of n into parts where if k is the largest part, all parts 1..k are present - Jon Perry (perry(AT)globalnet.co.uk), Sep 21 2005

a(n) = Sum(A117195(n,k): 0<=k<n) = A117192(n)+A117193(n) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 03 2006

The number of connected threshold graphs having n edges. - Michael D. Barrus (mbarrus2(AT)uiuc.edu), Jul 12 2007

Starting with offset 1 = row sums of triangle A146061 and the INVERT transform of A000700 starting: (1, 0, 1, -1, 1, -1, 1, -2, 2, -2, 2, -3, 3, -3, 4, -5,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 26 2008]

Number of partitions of n in which the largest part occurs an odd number of times and all other parts occur an even number of times. (Such partitions are the duals of the partitions with odd parts.) [From David Wasserman (dwasserm(AT)earthlink.net), Mar 04 2009]

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 11 2009: (Start)

Equals A035363 convolved with A010054. The convolution square of A000009 = A022567 = A000041 convolved with A010054.

A000041 = A000009 convolved with A035363. (End)

Considering all partitions of n into distinct parts: there are A140207(n) partitions of maximal size which is A003056(n), and A051162(n) is the greatest number occurring in these partitions. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 13 2009]

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 836.

Aicardi Francesca, MATRICIAL FORMULAE FOR PARTITIONS, arXiv:0806.1273.

G. E. Andrews, The Theory of Partitions, Cambridge University Press, 1998, p. 19.

G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.

R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; see p. 196.

T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, Problem 18.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 99.

W. Dunham, The Mathematical Universe, pp. 57-62 J.Wiley 1994.

Leonhard Euler, De partitione numerorum, Novi commentarii academiae scientiarum Petropolitanae 3 (1750/1), 1753, reprinted in: Commentationes Arithmeticae. (Opera Omnia. Series Prima: Opera Mathematica, Volumen Secundum), 1915, Lipsiae et Berolini, 254-294.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 277, Theorems 344, 346.

A. Lascoux, Sylvester's bijection between strict and odd partitions, Discrete Math., 277 (2004), 275-278.

C. J. Moreno and S. S. Wagstaff, Jr., Sums of Squares of Integers, Chapman and Hall, 2006, p. 253.

D. J. Newman, A Problem Seminar, pp. 18;93;102-3 Prob. 93 Springer-Verlag NY 1982.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..1999

Joerg Arndt, Fxtbook

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Author?, Sylvester's bijection

H. Bottomley, Illustration for A000009, A000041, A047967

Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions.

H. B. C. Darling, Collected Papers of Ramanujan, Table for q(n); n=1 through 100

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 48

E. Georgiadis, Computing Partition Numbers q(n), Technical Report, February (2009). [From Evangelos Georgiadis (egeorg(AT)mit.edu), Mar 19 2009]

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 108

J. Lovejoy, The Number Of Partitions Into Distinct Parts Modulo Powers Of 5

E. Sandifer, How Euler Did It, Philip Naude's problem

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (1)

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (2)

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (3)

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (4)

Wolfram Research, Generating functions for q(n)

Index entries for "core" sequences

G.f.: Product_{m >= 1} (1 + x^m) = 1/Product_{m >= 0} (1-x^(2m+1)) = Sum_{k>=0} Product_{i=1..k} x^i/(1-x^i).

Asymptotics: a(n) ~ exp(pi l_n / sqrt(3)) / ( 4 3^(1/4) l_n^(3/2) ) where l_n = (n-1/24)^(1/2) (Ayoub).

Product_{k=1..inf} (1+x^(2k)) = Sum_{k=0..inf} x^(k*(k+1))/Product_{i=1..k}(1-x^(2i)) - Euler (Hardy and Wright, Theorem 346).

For n>1, a(n)=(1/n)*Sum_{k=1..n} b(k)*a(n-k), with a(0)=1, b(n)= A000593(n)=sum of odd divisors of n; cf. A000700. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 21 2002

a(n) = t(n, 0), t as defined in A079211.

Given g.f. A(x), then B(x)=x*A(x^3)^8 satisfies 0=f(B(x), B(x^2)) where f(u, v)=v-u^2+16uv^2 . - Michael Somos May 31 2005

a(n)=A026837(n)+A026838(n)=A118301(n)+A118302(n); a(A001318(n))=A051044(n); a(A118300(n))=A118303(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 22 2006

Expansion of 1/chi(-q) in powers of q where chi() is a Ramanujan theta function. - Michael Somos May 28 2006

G.f. is Fourier series which satisfies f(-1/ (1152 t)) = 2^(-1/2)/ f(t) where q = exp(2 pi i t). - Michael Somos Aug 16 2007

Expansion of q^(-1/24) * eta(q^2) / eta(q) in powers of q.

Expansion of q^(-1/24) 2^(-1/2) f2(t) in powers of q = exp(2 Pi i t) where f2() is a Weber function. - Michael Somos Oct 18 2007

Given g.f. A(x), then B(x) = x * A(x^8)^3 satisfies 0 = f(B(x), B(x^3)) where f(u, v) = (u^3 - v) * (u + v^3) - 9 * u^3 * v^3. - Michael Somos, Mar 25 2008

Contribution from Evangelos Georgiadis, Andrew V. Sutherland, Kiran S. Kedlaya (egeorg(AT)mit.edu), Mar 03 2009: (Start)

a(0)=1. a(n)= 2*(Sum_{k=1} (-1)^(k+1) a(n-k^2)) + sigma(n) where

sigma(n)= (-1)^(j) if (n=(j*(3*j+1))/2 OR n=(j*(3*j-1))/2)

otherwise sigma(n)=0. (End)

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 13 2009: (Start)

The product G.f. = (1/(1-x))*(1/(1-x^3)*(1/(1-x^5)*...; = (1,1,1,...)*

(1,0,0,1,0,0,1,0,0,1,...)*(1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,...) * ...; =

a*b*c*... where a, a*b, a*b*c,...converge to A000009:

1, 1, 1, 2, 2, 2, 3, 3, 3,..4,... = a*b

1, 1, 1, 2, 2, 3, 4, 4, 5,..6,... = a*b*c

1, 1, 1, 2, 2, 3, 4, 5, 6,..7,... = a*b*c*d

1, 1, 1, 2, 2, 3, 4, 5, 6,..8,... = a*b*c*d*e

1, 1, 1, 2, 2, 3, 4, 5, 6,..8,... = a*b*c*d*e*f

...(Cf. analogous example in A000041). (End)

q + q^25 + q^49 + 2*q^73 + 2*q^97 + 3*q^121 + 4*q^145 + 5*q^169 +...

The partitions of n into distinct parts for small n are:

1: 1

2: 2

3: 3, 21

4: 4, 31

5: 5, 41, 32

6: 6, 51, 42, 321

7: 7, 61, 52, 43, 421

8: 8, 71, 62, 53, 521, 431

...

Contribution from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 13 2009: (Start)

a(8)=6, A140207(8)=#{5+2+1,4+3+1}=2, A003056(8)=3, A051162(8)=5;

a(9)=8, A140207(9)=#{6+2+1,5+3+1,4+3+2}=3, A003056(9)=3, A051162(9)=6;

a(10)=10, A140207(10)=#{4+3+2+1}=1, A003056(10)=4, A051162(10)=4. (End)

N := 100; t1 := series(mul(1+x^k, k=1..N), x, N); A000009 := proc(n) coeff(t1, x, n); end;

spec := [ P, {P=PowerSet(N), N=Sequence(Z, card>=1)} ]: [ seq(combstruct[count](spec, size=n), n=0..58) ];

spec := [ P, {P=PowerSet(N), N=Sequence(Z, card>=1)} ]: combstruct[allstructs](spec, size=10); # to get the actual partitions for n=10

A := mul(1+x^m, m=1..100); A000009 := n->coeff(A, x, n);

a[n_] := PartitionsQ[n]

PartitionsQ[Range[0, 60]] (from Harvey Dale, Jul 27 2009)

(PARI) a(n)=polcoeff(prod(k=1, n, 1+x^k, 1+x*O(x^n)), n)

(MAGMA) Coefficients(&*[1+x^m:m in [1..100]])[1..100] where x is PolynomialRing(Integers()).1; - from Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

(PARI) {a(n) = local(A); if(n<0, 0, A = x*O(x^n); polcoeff( eta(x^2+A)/ eta(x+A), n))}

Apart from the first term, equals A052839-1. The rows of A053632 converge to this sequence. When reduced modulo 2 equals the absolute values of A010815. The positions of odd terms given by A001318.

a(n)=sum(A097306(n, m), n=1..m), row sums of triangle of number of partitions of n into m odd parts.

Cf. A000041, A000700, A003724, A004111, A007837, A068049, A035294, A078408.

Cf. A081360, A088670, A109950, A109968, A132312.

Cf. A015723 (total number of parts).

A000700, A146061 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 26 2008]

Cf. A035363, A010054 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 11 2009]

Sequence in context: A034321 A058703 A034320 this_sequence A081360 A117409 A092833

Adjacent sequences: A000006 A000007 A000008 this_sequence A000010 A000011 A000012

nonn,core,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).