A006477 Number of partitions of n with at least 1 odd and 1 even part. (Formerly M3232)

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

0,6

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.

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

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 *)

Crossrefs

Cf. A047967, A038348.

Sequence in context: A188271 A219939 A219471 * A233739 A279036 A182699

Adjacent sequences: A006474 A006475 A006476 * A006478 A006479 A006480

Keyword

nonn

Author

N. J. A. Sloane.

Extensions

More terms from David W. Wilson, May 11 2001

Status

approved