A376513 Expansion of e.g.f. exp(x^3 * (1 + x)).
1, 0, 0, 6, 24, 0, 360, 5040, 20160, 60480, 1814400, 19958400, 99792000, 1037836800, 21794572800, 228843014400, 1743565824000, 29640619008000, 542423327846400, 6082255020441600, 70959641905152000, 1429329929803776000, 24977793950613504000
Offset: 0
Programs
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PARI
a(n) = n!*sum(k=0, n\3, binomial(k, n-3*k)/k!);
Formula
a(n) = n! * Sum_{k=0..floor(n/3)} binomial(k,n-3*k)/k!.
a(n) = (n-1)*(n-2) * (3*a(n-3) + 4*(n-3)*a(n-4)).
a(n) ~ 2^(n/2 - 1) * exp(-27/1024 + 45*2^(-19/2)*n^(1/4) - 9*n^(1/2)/64 + 2^(-3/2)*n^(3/4) - 3*n/4) * n^(3*n/4) * (1 + 264471/(5*2^(37/2)*n^(1/4))). - Vaclav Kotesovec, Sep 26 2024