A375683 Expansion of e.g.f. 1 / (1 + x * (exp(x) - 1)).

1, 0, -2, -3, 20, 115, -306, -6307, -6616, 462663, 2956130, -38945951, -656504388, 2325876683, 145820995670, 562691968005, -33452317341616, -449954883966065, 7055017491780810, 233802046526955497, -571834988279277340, -112474674691684827501

Offset: 0

Author

Seiichi Manyama, Aug 24 2024

Keywords

• sign

Crossrefs

Cf. A052848, A367880, A367881.

Cf. A292893, A375684.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x*(exp(x)-1))))

PARI

a(n) = n!*sum(k=0, n\2, (-1)^k*k!*stirling(n-k, k, 2)/(n-k)!);

Formula

a(0) = 1; a(n) = -n * Sum_{k=2..n} binomial(n-1,k-1) * a(n-k).

a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * k! * Stirling2(n-k,k)/(n-k)!.