A376512 Expansion of e.g.f. exp(x^2 * (1 + x)).
1, 0, 2, 6, 12, 120, 480, 2520, 21840, 120960, 937440, 8316000, 60540480, 570810240, 5465940480, 49037788800, 523588665600, 5504686387200, 57816850291200, 678823104960000, 7844848544332800, 93064133530368000, 1184800751111577600, 14967781957781452800
Offset: 0
Programs
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PARI
a(n) = n!*sum(k=0, n\2, binomial(k, n-2*k)/k!);
Formula
a(n) = n! * Sum_{k=0..floor(n/2)} binomial(k,n-2*k)/k!.
a(n) = (n-1) * (2*a(n-2) + 3*(n-2)*a(n-3)).
a(n) ~ 3^(n/3 - 1/2) * exp(4/81 - 2*3^(-7/3)*n^(1/3) + 3^(-2/3)*n^(2/3) - 2*n/3) * n^(2*n/3) * (1 + 223/(3^(20/3)*n^(1/3))). - Vaclav Kotesovec, Sep 26 2024