A382349 a(n) = [x^n] Product_{k=0..n} (1 + (3*n+k)*x).

1, 7, 146, 4578, 189144, 9660840, 586813968, 41283943344, 3299858098560, 295294500123840, 29242449106502400, 3174506423754019200, 374845813851886709760, 47828682507084551654400, 6557612642418946942310400, 961431335221085133398784000, 150095351600371197275428454400

Offset: 0

Author

Seiichi Manyama, May 18 2025

Keywords

• nonn

Crossrefs

Column k=3 of A382347.

Cf. A165675.

Programs

Mathematica

Table[SeriesCoefficient[Product[(1 + (3*n+k)*x), {k, 0, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 23 2025 *)

PARI

a(n) = sum(k=0, n, (k+1)*(3*n)^k*abs(stirling(n+1, k+1, 1)));

Formula

a(n) = A165675(4*n,3*n).

a(n) = Sum_{k=0..n} (k+1) * (3*n)^k * |Stirling1(n+1,k+1)|.

a(n) = (n+1)! * Sum_{k=0..n} (-1)^k * binomial(-3*n,k)/(n+1-k).

a(n) = (4*n)!/(3*n)! * (1 + 3*n * Sum_{k=1..n} 1/(3*n+k)).

a(n) ~ log(4/3) * 2^(8*n+1) * n^(n+1) / (exp(n) * 3^(3*n - 1/2)). - Vaclav Kotesovec, May 23 2025