A354413 Expansion of e.g.f. 1/(2 - exp(x))^x.
1, 0, 2, 6, 36, 250, 2100, 20594, 231168, 2923722, 41149140, 637972522, 10804678632, 198480649250, 3930963078588, 83500876635570, 1893745346613216, 45672635292831322, 1167233799092342148, 31510575263852229242, 896028017040096045720
Offset: 0
Keywords
Programs
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Mathematica
With[{nn=20},CoefficientList[Series[1/(2-Exp[x])^x,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 03 2025 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(x))^x)) -
PARI
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j*sum(k=1, j-1, (k-1)!*stirling(j-1, k, 2))*binomial(i-1, j-1)*v[i-j+1])); v;
Formula
a(0) = 1; a(n) = Sum_{k=1..n} A052862(k) * binomial(n-1,k-1) * a(n-k).
a(n) ~ n! / (Gamma(log(2)) * 2^log(2) * n^(1 - log(2)) * log(2)^(n + log(2))). - Vaclav Kotesovec, Jun 08 2022