A362857 Expansion of e.g.f. exp(-2*x) / (1 + LambertW(-x)).

1, -1, 4, 7, 120, 1373, 21028, 373931, 7670736, 178064281, 4615519884, 132139421423, 4141235867992, 141016013784917, 5184372688776180, 204668397165154867, 8635388122600110240, 387787185320578895537, 18467131524896950511644

Offset: 0

Author

Seiichi Manyama, May 05 2023

Keywords

• sign

Links

• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Column k=2 of A362856.

Cf. A362859.

Programs

Mathematica

With[{nn=20},CoefficientList[Series[Exp[-2x]/(1+LambertW[-x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 26 2023 *)

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-2*x)/(1 + lambertw(-x))))

Formula

G.f.: Sum_{k>=0} (k*x)^k / (1 + 2*x)^(k+1).

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

a(n) ~ exp(-2*exp(-1)) * n^n. - Vaclav Kotesovec, Aug 05 2025