A335813 Expansion of e.g.f. Product_{k>=1} (1 + (1 - exp(x))^k).

1, -1, 1, -7, -11, -151, -419, -1807, -5291, -381031, -9125939, -139879807, -1217973371, 7055720489, 657464911741, 20268419534993, 455079458957749, 7487596915540409, 62151133224856621, -943454812059725407, -32387452121872219931, 1120264679544729734729

Offset: 0

Author

Ilya Gutkovskiy, Jun 25 2020

Keywords

• sign

Comments

Inverse binomial transform of A335811.

Crossrefs

Cf. A000009, A305550, A335811, A335812.

Programs

Mathematica

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

PARI

N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1+(1-exp(x))^k))) \\ Seiichi Manyama, Jul 08 2020

Formula

a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * k! * A000009(k).