A055922 Number of partitions of n in which each part occurs an odd number (or zero) times.

1, 1, 1, 3, 2, 5, 6, 9, 9, 16, 20, 25, 32, 40, 54, 69, 84, 101, 136, 156, 202, 244, 306, 357, 448, 527, 652, 773, 944, 1103, 1346, 1574, 1885, 2228, 2640, 3106, 3684, 4302, 5052, 5931, 6924, 8079, 9416, 10958, 12718, 14824, 17078, 19820, 22860, 26433

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

F. C. Auluck, K. S. Singwi and B. K. Agarwala, On a new type of partition, Proc. Nat. Inst. Sci. India 16 (1950) 147-156.

Steven Finch, Integer Partitions, September 22, 2004, page 5. [Cached copy, with permission of the author]

Daniel Herden, Mark R. Sepanski, Jonathan Stanfill, Cordell Hammon, Joel Henningsen, Henry Ickes, and Indalecio Ruiz, Partitions With Designated Summands Not Divisible by 2^L, 2, and 3^L Modulo 2, 4, and 3, arXiv:2101.04058 [math.CO], 2021. See also Integers (2023) Vol. 23, Art. No. A43.

James A. Sellers and Fabrizio Zanello, On the parity of the number of partitions with odd multiplicities, arXiv:2004.06204 [math.CO], 2020.

Formula

EULER transform of b where b has g.f. Sum {k>0} c(k)*x^k/(1-x^k) where c is inverse EULER transform of characteristic function of odd numbers.

G.f.: Product_{i>0} (1+x^i-x^(2*i))/(1-x^(2*i)). - Vladeta Jovovic, Feb 03 2004

Asymptotics (Auluck, Singwi, Agarwala, 1950): a(n) ~ B/(2*Pi*n) * exp(2*B*sqrt(n)), where B = sqrt(Pi^2/12 + 2*log(phi)^2), where phi is the golden ratio. - Vaclav Kotesovec, Oct 27 2014

Example

There exist 11 partitions of 6. For six of these partitions, each part occurs an odd number times, they are 6 = 5 + 1 = 4 + 2 = 3 + 2 + 1 = 3 + 1+1+1 = 2+2+2, hence a(6) = 6. The five other partitions are 4 + 1+1 = 3+3 = 2+2 + 1+1 = 2 + 1+1+1+1 = 1+1+1+1+1+1.

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(j, 2)=0, 0, b(n-i*j, i-1)), j=1..n/i)
       +b(n, i-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 31 2014

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[Mod[j, 2] == 0, 0, b[n-i*j, i-1]], {j, 1, n/i}] + b[n, i-1]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)

Prog

(PARI) { my(n=60); Vec(prod(k=1, n, 1 + sum(r=0, n\(2*k), x^(k*(2*r+1))) + O(x*x^n))) } \\ Andrew Howroyd, Dec 22 2017

Crossrefs

Cf. A000041, A007690, A055923, A249389.

Column k=0 of A264399.

Sequence in context: A303764 A079973 A267150 * A194072 A194105 A194012

Adjacent sequences: A055919 A055920 A055921 * A055923 A055924 A055925

Keyword

nonn

Author

Christian G. Bower, Jun 23 2000

Status

approved