A328008 Expansion of e.g.f. 1 / (2 - exp(x) / (1 - x)).

1, 2, 13, 124, 1575, 25006, 476421, 10589720, 269010979, 7687905826, 244120131393, 8526912775756, 324914136199263, 13412430958497494, 596253684006657085, 28399969571266895488, 1442890578572155475355, 77889310498718258171914, 4451905168738601015593785

Offset: 0

Author

Ilya Gutkovskiy, Oct 01 2019

Keywords

• nonn

Crossrefs

Cf. A000522, A002866, A321974, A328007.

Programs

Mathematica

nmax = 18; CoefficientList[Series[1/(2 - Exp[x]/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Floor[Exp[1] k!] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

PARI

my(x='x+O('x^25)); Vec(serlaplace(1/(2-exp(x)/(1-x)))) \\ Michel Marcus, Oct 02 2019

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A000522(k) * a(n-k).

a(n) ~ n! / (2*(1 + 1/LambertW(exp(1)/2)) * (1 - LambertW(exp(1)/2))^(n+1)). - Vaclav Kotesovec, Oct 02 2019