A362892 Expansion of e.g.f. 1/(1 + LambertW(-x^2 * (exp(x) - 1))).
1, 0, 0, 6, 12, 20, 1470, 10122, 47096, 1814472, 25119450, 226527950, 6732015972, 142901684796, 2071229736758, 57596022404130, 1589579741044080, 32832196825559312, 951335638952843826, 31043287459520549910, 838738470701197009820
Offset: 0
Keywords
Links
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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Mathematica
With[{nn=20},CoefficientList[Series[1/(1+LambertW[-x^2(Exp[x]-1)]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 14 2025 *) -
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-x^2*(exp(x)-1)))))
Formula
a(n) = n! * Sum_{k=0..floor(n/3)} k^k * Stirling2(n-2*k,k)/(n-2*k)!.
a(n) ~ n^n / (sqrt((2+r)*exp(r) - 2) * r^(n+1) * exp(n + 1/2)), where r = 0.640353588740603511543638690178204955926349... is the root of the equation r^2*(exp(r+1) - exp(1)) = 1. - Vaclav Kotesovec, May 19 2025