A383384 Expansion of e.g.f. exp(-5*x) / (1-x)^5.
1, 0, 5, 10, 105, 620, 5725, 52950, 571025, 6686200, 85871925, 1193029250, 17846277625, 285737086500, 4874590170125, 88245858436750, 1689282139310625, 34088182903910000, 723088091207873125, 16083522103093616250, 374280288623526655625, 9093957982779894737500
Offset: 0
Programs
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PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-5*x)/(1-x)^5))
Formula
a(n) = n! * Sum_{k=0..n} (-5)^(n-k) * binomial(k+4,4)/(n-k)!.
a(n) = (n-1) * (a(n-1) + 5*a(n-2)) for n > 1.
E.g.f.: B(x)^5, where B(x) is the e.g.f. of A000166.
a(n) ~ sqrt(2*Pi) * n^(n + 9/2) / (24*exp(n+5)). - Vaclav Kotesovec, Apr 25 2025