A006477 Number of partitions of n with at least 1 odd and 1 even part.

0, 0, 0, 1, 1, 4, 4, 10, 11, 22, 25, 44, 51, 83, 98, 149, 177, 259, 309, 436, 521, 716, 857, 1151, 1376, 1816, 2170, 2818, 3361, 4309, 5132, 6502, 7728, 9695, 11501, 14298, 16924, 20877, 24661, 30203, 35598, 43323, 50956, 61651, 72357, 87086, 101999

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• M. O. LeVan, A triangle for partitions, Amer. Math. Monthly, 79 (1972), 507-510.

Crossrefs

Cf. A047967, A038348.

Programs

Mathematica

a[n_?OddQ] := PartitionsP[n] - PartitionsQ[n]; a[n_?EvenQ] := PartitionsP[n] - PartitionsQ[n] - PartitionsP[n/2]; a[0] = 0; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 17 2014, after Vladeta Jovovic *)

Formula

Convolution of 0, 1, 1, 2, 2, 3, 4, 5, 6, ... (essentially A000009) and 0, 0, 1, 0, 2, 0, 3, 0, 5, ... (essentially A035363).

G.f.: (prod(1/(1-x^k), k odd)-1) * (prod(1/(1-x^k), k even)-1).

A000041(n)-A000009(n) if n is odd else A000041(n)-A000009(n)-A000041(n/2). - Vladeta Jovovic, Sep 10 2003

a(n) = A000041(n) - A096441(n), n >= 1. - Omar E. Pol, Aug 16 2013

Extensions

More terms from David W. Wilson, May 11 2001