A380707 a(n) = [x^n] Product_{k=0..n} (1 + (n^2+k)*x).

1, 3, 74, 4578, 520024, 93638820, 24469489008, 8744195444880, 4093736159733120, 2430707964048640800, 1784480276787736636800, 1586934417435493101528960, 1680937045347184025188838400, 2091005717306225140393765228800, 3018259634660179964662904164915200

Offset: 0

Author

Seiichi Manyama, May 18 2025

Keywords

• nonn

Crossrefs

Main diagonal of A382347.

Cf. A165675.

Programs

Mathematica

Table[SeriesCoefficient[Product[1 + (n^2+k)*x, {k, 0, n}], {x, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, May 23 2025 *)

PARI

a(n) = sum(k=0, n, (k+1)*n^(2*k)*abs(stirling(n+1, k+1, 1)));

Formula

a(n) = A165675((n+1)*n,n^2).

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

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

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

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