A305990 Expansion of e.g.f.: (1+x) / (exp(-x) - x).

1, 3, 11, 58, 409, 3606, 38149, 470856, 6641793, 105398650, 1858413061, 36044759796, 762659322385, 17481598316742, 431535346662645, 11413394655983536, 321989729198400385, 9651573930139850610, 306321759739045148293, 10262156907184058219340

Offset: 0

Author

Vaclav Kotesovec, Jun 16 2018

Keywords

• nonn

Crossrefs

Cf. A305133, A006153, A009306, A009444, A072597, A089148, A302397.

Programs

Mathematica

nmax = 20; CoefficientList[Series[(1+x)/(E^(-x)-x), {x, 0, nmax}], x] * Range[0, nmax]!
a={1};For[n=1,n<20,n++,AppendTo[a,Sum[(n!)*((n-k+1)^(k-1))*(n+1)/(k!),{k,0,n+1}]]]; a (* Detlef Meya, Sep 05 2023 *)

Formula

a(n) ~ n! / LambertW(1)^(n+1).

a(n) = (-1)^n * A009444(n+1).

a(n) = Sum_{k=0..n+1} (n+1)!*(n-k+1)^(k-1)/k! for n > 0. - Detlef Meya, Sep 05 2023