A298596 Expansion of Product_{k>=2} 1/(1 + x^k).

1, 0, -1, -1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, -1, 1, 0, 0, -1, 1, -1, 0, -1, 2, -1, 0, -2, 2, -1, 1, -2, 3, -2, 1, -3, 4, -2, 2, -4, 5, -3, 3, -5, 6, -5, 4, -6, 9, -6, 5, -9, 10, -8, 8, -11, 13, -11, 10, -14, 17, -14, 13, -19, 21, -18, 18, -23, 26, -24, 23, -29, 34, -30, 29, -38, 41, -38, 39

Offset: 0

Author

Ilya Gutkovskiy, Jan 22 2018

Keywords

• sign

Comments

The difference between the number of partitions of n into an even number of parts > 1 and the number of partitions of n into an odd number of parts > 1.

Convolution inverse of A025147.

Links

• Index entries for related partition-counting sequences

Crossrefs

Cf. A000700, A002865, A025147, A027349, A078616, A081362.

Programs

Mathematica

nmax = 78; CoefficientList[Series[Product[1/(1 + x^k), {k, 2, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=2} 1/(1 + x^k).

a(n) = (-1)^n * (A000700(n) - A000700(n-1)), for n > 0. - Vaclav Kotesovec, Jun 06 2018

a(n) ~ (-1)^n * Pi * exp(Pi*sqrt(n/6)) / (2^(13/4) * 3^(3/4) * n^(5/4)). - Vaclav Kotesovec, Jun 06 2018

a(n) = A027188(n+2) - A027194(n+2). - R. J. Mathar, Jun 16 2018