A309775 Expansion of e.g.f. exp(2 * (1 - exp(x)) + x).

1, -1, -1, 3, 7, -13, -89, -45, 1191, 4723, -6873, -143597, -499289, 1843891, 28132391, 104223059, -508838745, -8597456141, -39770287321, 158845792147, 3788893515687, 23979078221619, -38626203043289, -2200108609291821, -19878849864738137, -27269435066568845

Offset: 0

Author

Seiichi Manyama, Jul 06 2020

Keywords

• sign

Crossrefs

Column k=2 of A335977.

Cf. A335980.

Programs

Mathematica

m = 25; Range[0, m]! * CoefficientList[Series[Exp[2 * (1 - Exp[x]) + x], {x, 0, m}], x] (* Amiram Eldar, Jul 06 2020 *)
Table[Sum[Binomial[n, k] * BellB[k, -2], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Jul 06 2020 *)

PARI

N=40; x='x+O('x^N); Vec(serlaplace(exp(2*(1-exp(x))+x)))

Formula

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

a(n) = exp(2) * Sum_{k>=0} (k + 1)^n * (-2)^k / k!.

a(n) = Sum_{k=0..n} binomial(n,k) * Bell(k, -2). - Vaclav Kotesovec, Jul 06 2020