A327601 Expansion of e.g.f. exp(x) * Product_{k>=1} 1/(1 - (1 - exp(x))^k).

1, 0, 2, 0, 26, 120, 1922, 21840, 307946, 4251240, 63165842, 1010729280, 18501318266, 391496665560, 9265945721762, 232411950454320, 5972325812958986, 156131611764907080, 4208451299935189682, 119669466221148348960, 3658459009408581118106

Offset: 0

Author

Ilya Gutkovskiy, Sep 18 2019

Keywords

• nonn

Comments

Stirling-Bernoulli transform of partition numbers (A000041).

Crossrefs

Cf. A000041, A167137, A306022, A306042.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Exp[x] Product[1/(1 - (1 - Exp[x])^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k StirlingS2[n + 1, k + 1] k! PartitionsP[k], {k, 0, n}], {n, 0, 20}]

PARI

a(n) = sum(k=0, n, (-1)^k*stirling(n+1, k+1, 2)*k!*numbpart(k)); \\ Michel Marcus, Sep 19 2019

Formula

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