A383344 Expansion of e.g.f. exp(-4*x) / (1-x)^4.
1, 0, 4, 8, 72, 416, 3520, 31104, 316288, 3525632, 43117056, 572195840, 8191304704, 125761056768, 2060841582592, 35894401335296, 662066514984960, 12890305925218304, 264155723747688448, 5682905054074109952, 128051031032232411136, 3015653024970577018880
Offset: 0
Programs
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PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-4*x)/(1-x)^4))
Formula
a(n) = n! * Sum_{k=0..n} (-4)^(n-k) * binomial(k+3,3)/(n-k)!.
a(n) = (n-1) * (a(n-1) + 4*a(n-2)) for n > 1.
E.g.f.: B(x)^4, where B(x) is the e.g.f. of A000166.
a(n) ~ sqrt(2*Pi) * n^(n + 7/2) / (6*exp(n+4)). - Vaclav Kotesovec, Apr 25 2025