A335812 E.g.f.: Product_{k>=1} 1 / (1 - (1 - exp(x))^k).

1, -1, 3, -7, 39, -31, 1623, 9953, 182199, 2116289, 32269143, 505278113, 9743069559, 214428606209, 5156280298263, 127321200213473, 3176128419544119, 80737907621585729, 2147513299611040983, 61423058495936864033, 1912348969322283717879, 64216042408215934910849

Offset: 0

Author

Ilya Gutkovskiy, Jun 25 2020

Keywords

• sign

Comments

Inverse binomial transform of A327601.

Crossrefs

Cf. A000041, A167137, A327601, A335813.

Programs

Mathematica

nmax = 21; CoefficientList[Series[ Product[1/(1 - (1 - Exp[x])^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k StirlingS2[n, k] k! PartitionsP[k], {k, 0, n}], {n, 0, 21}]

PARI

a(n) = sum(k=0, n, (-1)^k * stirling(n,k,2) * k! * numbpart(k)); \\ Michel Marcus, Jun 25 2020

Formula

a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * k! * A000041(k).