A335945 E.g.f. A(x) satisfies A(x) = exp(x*A(x)/(1 + x)).

1, 1, 1, 4, 17, 116, 907, 9010, 102097, 1348408, 19939571, 330204854, 6015657529, 120016789348, 2597201945899, 60667591974826, 1520434054966433, 40710815980598000, 1159627208850209251, 35018022339726428926, 1117395892399939407241, 37569709612314269554396

Offset: 0

Author

Ilya Gutkovskiy, Jul 01 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..414
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A052868, A060356, A108919.

Programs

Mathematica

nmax = 21; A[] = 0; Do[A[x] = Exp[x A[x]/(1 + x)] + O[x]^(nmax + 1) // Normal, nmax + 1];CoefficientList[A[x], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[-(1 + x) LambertW[-x/(1 + x)]/x, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] (k + 1)^(k - 1) n!/k!, {k, 0, n}], {n, 0, 21}]

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1+x))))) \\ Seiichi Manyama, Mar 05 2023

Formula

E.g.f.: -(1 + x) * LambertW(-x/(1 + x)) / x.

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * (k+1)^(k-1) * n! / k!.

a(n) ~ (exp(1) - 1)^(n + 1/2) * n^(n-1) / exp(n - 1/2). - Vaclav Kotesovec, Jul 01 2020

E.g.f.: exp ( -LambertW(-x/(1+x)) ). - Seiichi Manyama, Mar 05 2023