A309775 - OEIS

%I #60 Jul 06 2020 14:54:08

%S 1,-1,-1,3,7,-13,-89,-45,1191,4723,-6873,-143597,-499289,1843891,

%T 28132391,104223059,-508838745,-8597456141,-39770287321,158845792147,

%U 3788893515687,23979078221619,-38626203043289,-2200108609291821,-19878849864738137,-27269435066568845

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

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

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

%F a(n) = Sum_{k=0..n} binomial(n,k) * Bell(k, -2). - _Vaclav Kotesovec_, Jul 06 2020

%t m = 25; Range[0, m]! * CoefficientList[Series[Exp[2 * (1 - Exp[x]) + x], {x, 0, m}], x] (* _Amiram Eldar_, Jul 06 2020 *)

%t Table[Sum[Binomial[n, k] * BellB[k, -2], {k, 0, n}], {n, 0, 30}] (* _Vaclav Kotesovec_, Jul 06 2020 *)

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

%Y Column k=2 of A335977.

%Y Cf. A335980.

%K sign

%O 0,4

%A _Seiichi Manyama_, Jul 06 2020