A351507 a(n) = [x^n] Product_{k=1..n} (1 + k*x)^n.

1, 1, 13, 630, 65446, 11732175, 3222746276, 1257489289280, 661150527657180, 450529025613124095, 386183537239831846450, 406654820487854059966416, 516014868401631381045209376, 776565429016998902169538130936, 1367544537916924083498367095477800

Offset: 0

Author

Seiichi Manyama, Feb 12 2022

Keywords

• nonn

Crossrefs

Cf. A000142, A129256, A384012, A384031.

Cf. A351508, A384017.

Programs

Mathematica

Table[SeriesCoefficient[Product[(1 + k*x)^n, {k,1,n}], {x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Feb 18 2022 *)

PARI

a(n) = polcoef(prod(k=1, n, 1+k*x)^n, n);

Formula

a(n) ~ exp(n + 1/3) * n^(2*n - 1/2) / (sqrt(Pi) * 2^(n + 1/2)). - Vaclav Kotesovec, Feb 18 2022

a(n) = Sum_{0 <= x_1, x_2,..., x_n <= n and x_1 + x_2 + ... + x_n = (n-1)*n} Product_{k=1..n} |Stirling1(n + 1,x_k + 1)|. - Seiichi Manyama, May 18 2025