A373741 Expansion of e.g.f. exp(x^2/2 * (1 + x)^3).
1, 0, 1, 9, 39, 150, 1365, 13545, 105945, 918540, 10603845, 127806525, 1468823895, 18253765530, 257397445305, 3770163121725, 55637459903025, 866703333295800, 14468243658093225, 250223925107581425, 4426399346291497575, 81488489549760042750
Offset: 0
Keywords
Programs
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Mathematica
With[{nn=30},CoefficientList[Series[Exp[x^2/2 (1+x)^3],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 26 2025 *) -
PARI
a(n) = n!*sum(k=0, n\2, binomial(3*k, n-2*k)/(2^k*k!));
Formula
a(n) = n! * Sum_{k=0..floor(n/2)} binomial(3*k,n-2*k)/(2^k * k!).
a(n) = (n-1)/2 * (2*a(n-2) + 9*(n-2)*a(n-3) + 12*(n-2)*(n-3)*a(n-4) + 5*(n-2)*(n-3)*(n-4)*a(n-5)).