A383678 a(n) = [x^n] Product_{k=0..n} (1 + (2*n+k)*x).

1, 5, 74, 1650, 48504, 1763100, 76223664, 3817038960, 217177416576, 13834411290720, 975244141065600, 75366122480858880, 6335159176892851200, 575442172080117538560, 56165570794932257433600, 5862137958472255891200000, 651508569509254106827161600, 76814449419352043102473728000

Offset: 0

Author

Seiichi Manyama, May 18 2025

Keywords

• nonn

Crossrefs

Column k=2 of A382347.

Cf. A000142, A165675, A384024.

Programs

Mathematica

Table[SeriesCoefficient[Product[(1 + (2*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)*(2*n)^k*abs(stirling(n+1, k+1, 1)));

Formula

a(n) = A165675(3*n,2*n).

a(n) = Sum_{k=0..n} (k+1) * (2*n)^k * |Stirling1(n+1,k+1)|.

a(n) = (n+1)! * Sum_{k=0..n} (-1)^k * binomial(-2*n,k)/(n+1-k).

a(n) = (3*n)!/(2*n)! * (1 + 2*n * Sum_{k=1..n} 1/(2*n+k)).

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