A354624 Expansion of e.g.f. (1 - x)^(-x^4).
1, 0, 0, 0, 0, 120, 360, 1680, 10080, 72576, 2419200, 25660800, 279417600, 3286483200, 41894012160, 794511244800, 13755488947200, 238514695372800, 4269265386946560, 79696849513881600, 1658065431859200000
Offset: 0
Keywords
Programs
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PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-x^4))) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x^4*log(1-x)))) -
PARI
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=5, i, j/(j-4)*v[i-j+1]/(i-j)!)); v;
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PARI
a(n) = n!*sum(k=0, n\5, abs(stirling(n-4*k, k, 1))/(n-4*k)!);
Formula
a(0) = 1; a(n) = (n-1)! * Sum_{k=5..n} k/(k-4) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/5)} |Stirling1(n-4*k,k)|/(n-4*k)!.
a(n) ~ n! * (1 - 4/n - 16*log(n)/n^2). - Vaclav Kotesovec, Jul 21 2022