A354412 Expansion of e.g.f. 1/(2 - exp(x))^(x/2).
1, 0, 1, 3, 15, 95, 735, 6727, 71169, 854919, 11497845, 171179261, 2795081751, 49668211177, 954226247247, 19709181213555, 435524370171393, 10252531220906051, 256148413939459917, 6769302493147288885, 188664988853982963735, 5530544750788380455433
Offset: 0
Keywords
Programs
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Mathematica
With[{nn=30},CoefficientList[Series[1/(2-Exp[x])^(x/2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 12 2024 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(x))^(x/2))) -
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])/2); v;
Formula
a(0) = 1; a(n) = (1/2) * Sum_{k=1..n} A052862(k) * binomial(n-1,k-1) * a(n-k).
a(n) ~ n! / (Gamma(log(2)/2) * 2^(log(2)/2) * n^(1 - log(2)/2) * log(2)^(n + log(2)/2)). - Vaclav Kotesovec, Jun 08 2022