A277509 Expansion of e.g.f. 1/((1+LambertW(-x))*(1+x)).
1, 0, 4, 15, 196, 2145, 33786, 587041, 12080888, 278692497, 7213075030, 205967845281, 6444486304884, 219096784628761, 8044651840755362, 317224112769528945, 13371158269397088496, 599930571306586259745, 28547657791777984900014, 1436014157616531876023713
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..385
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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Mathematica
CoefficientList[Series[1/(1+LambertW[-x])/(1+x), {x, 0, 20}], x] * Range[0, 20]! Flatten[{1, Table[(-1)^n*n! + Sum[(-1)^(n-k) * Binomial[n, k] * k^k * (n-k)!, {k, 1, n}], {n, 1, 20}]}] -
PARI
my(x='x+O('x^50)); Vec(serlaplace(1/((1 + lambertw(-x))*(1+x)))) \\ G. C. Greubel, Nov 12 2017
Formula
For n > 0, a(n) = (-1)^n*n!+Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * k^k * (n-k)!.
a(n) ~ n^n / (1+exp(-1)).
a(0) = 1; a(n) = -n*a(n-1) + n^n. - Seiichi Manyama, May 01 2023