A336097 E.g.f.: Product_{k>=1} (1 - (1 - exp(x))^k).

1, 1, -1, -5, -13, -149, -1861, -21965, -267373, -3163109, -34739221, -352104125, -3806609533, -67068890069, -1866226978981, -51776974365485, -1180415240484493, -19613026052409029, -122604194898649141, 6950364605049945955, 394565422299921179747, 13840685990526765512011

Offset: 0

Author

Seiichi Manyama, Jul 08 2020

Keywords

• sign

Crossrefs

Cf. A010815, A167137, A335812, A335813, A336100.

Programs

Mathematica

m = 21; Range[0, m]! * CoefficientList[Series[Product[1 - (1 - Exp[x])^k, {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, Jul 08 2020 *)
A010815[k_] := (m = (1 + Sqrt[1 + 24*k])/6; If[IntegerQ[m], (-1)^m, 0] + If[IntegerQ[m - 1/3], (-1)^(m - 1/3), 0]); Table[Sum[(-1)^k * StirlingS2[n, k] * k! * A010815[k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 09 2020 *)

PARI

N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1-(1-exp(x))^k)))

Formula

a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * k! * A010815(k).