A354317 Expansion of e.g.f. exp(-log(1 + x)^2 / 2).
1, 0, -1, 3, -8, 20, -34, -126, 2514, -28008, 285774, -2922810, 30858048, -339954264, 3920819748, -47319853140, 596005041852, -7799132781792, 105344546511684, -1454910026870412, 20242465245436128, -276289562032117200, 3490199850169557480
Offset: 0
Keywords
Programs
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PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-log(1+x)^2/2))) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x)^(log(1+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, binomial(i-1, j-1)*stirling(j, 2, 1)*v[i-j+1])); v;
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PARI
a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 1)/((-2)^k*k!));
Formula
E.g.f.: 1/(1 + x)^(log(1 + x)/2).
a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n-1,k-1) * Stirling1(k,2) * a(n-k).
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * Stirling1(n,2*k)/((-2)^k * k!).