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

Author

Christian G. Bower, Jun 23 2000

Keywords

• nonn

Examples

			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.
		

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.

Crossrefs

Cf. A000041, A007690, A055923, A249389.

Column k=0 of A264399.

Programs

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 *)

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

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