A354389 Expansion of e.g.f. 1/(1 + log(1 + x)^2 / 2).
1, 0, -1, 3, -5, -10, 146, -756, 1086, 23400, -300066, 1855590, 341826, -165915828, 2158958556, -10006622640, -172337345496, 4941605486016, -64365944851512, 339328464492456, 5510899593823176, -157099566384759600, 1059259019507498160, 41957473280879898720
Offset: 0
Keywords
Programs
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Mathematica
With[{nn=30},CoefficientList[Series[1/(1+(Log[1+x]^2)/2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 08 2024 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1+x)^2/2))) -
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, j)*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);
Formula
a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * Stirling1(k,2) * a(n-k).
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * Stirling1(n,2*k)/(-2)^k.