A304784 Expansion of Product_{k>=1} 1/(1 + x^k)^p(k), where p(k) = number of partitions of k (A000041).

1, -1, -1, -2, 0, -1, 2, 3, 11, 8, 19, 13, 22, -5, -10, -80, -105, -246, -303, -502, -506, -681, -400, -231, 873, 1956, 4733, 7536, 12891, 17609, 25188, 29508, 34890, 29690, 19039, -17742, -74002, -183563, -333665, -572271, -866683, -1271429, -1698491, -2181207

Offset: 0

Author

Ilya Gutkovskiy, May 18 2018

Keywords

• sign

Comments

Convolution inverse of A261049.

Links

• Eric Weisstein's World of Mathematics, Partition Function P
• Index entries for sequences related to partitions

Crossrefs

Cf. A000041, A001970, A089254, A261049, A300508.

Programs

Mathematica

nmax = 43; CoefficientList[Series[Product[1/(1 + x^k)^PartitionsP[k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d) d PartitionsP[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 43}]

Formula

G.f.: Product_{k>=1} 1/(1 + x^k)^A000041(k).