A383221 Coefficient of x^3 in expansion of (x+2) * (x+5) * ... * (x+3*n-1).

0, 0, 0, 1, 26, 595, 14155, 363944, 10206700, 312193524, 10380710220, 373619597736, 14490750497432, 603032132116336, 26818416624389936, 1269883590624201344, 63806666669904903808, 3391580011320726010880, 190174443042558311293440, 11220246602286014617751040

Offset: 0

Author

Seiichi Manyama, Apr 20 2025

Keywords

• nonn

Crossrefs

Column k=3 of A225470.

Programs

PARI

a(n) = polcoef(prod(k=0, n-1, x+3*k+2), 3);

Formula

a(n) = Sum_{k=3..n} 2^(k-3) * 3^(n-k) * binomial(k,3) * |Stirling1(n,k)|.

E.g.f.: f(x)^2 * log(f(x))^3 / 6, where f(x) = 1/(1 - 3*x)^(1/3).

a(n) = Sum_{k=3..n} (3*n-1)^(k-3) * 3^(n-k) * binomial(k,3) * Stirling1(n,k). - Seiichi Manyama, May 06 2025