A354610 Expansion of e.g.f. exp(f(x) - 1) where f(x) = (1 - x)^x = e.g.f. for A007114.
1, 0, -2, -3, 16, 90, -84, -2940, -8672, 95256, 956160, -811800, -75724296, -419150160, 4406562720, 78306555600, 89704074240, -9655388184960, -97621097227200, 657339885653760, 23680733504400000, 119677890314505600, -3528587069869276800, -64401874868363598720
Offset: 0
Keywords
Programs
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PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((1-x)^x-1))) -
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=0, j\2, (-1)^(j-k)*stirling(j-k, k, 1)/(j-k)!)*binomial(i-1, j-1)*v[i-j+1])); v;
Formula
a(0) = 1; a(n) = Sum_{k=1..n} A007114(k) * binomial(n-1,k-1) * a(n-k).