A384022 a(n) = [x^(2*n)] Product_{k=0..n} 1/(1 - k*x)^3.

1, 6, 699, 242434, 170580831, 202617635850, 364680579642546, 926271490234962816, 3156974021179142865351, 13905988122027295313489800, 76896867190774672671251191752, 521595538342870729288480053506382, 4258687803431080424982372253063299050, 41202042785933045982333959380025893914894

Offset: 0

Author

Seiichi Manyama, May 17 2025

Keywords

• nonn

Crossrefs

Cf. A383862.

Programs

Mathematica

Table[SeriesCoefficient[Product[1/(1-k*x)^3, {k, 1, n}], {x, 0, 2*n}], {n, 0, 15}] (* Vaclav Kotesovec, May 18 2025 *)
(* or *)
Table[Sum[StirlingS2[i+n, n] * StirlingS2[j+n, n] * StirlingS2[3*n-i-j, n], {i, 0, 2*n}, {j, 0, 2*n-i}], {n, 0, 15}] (* Vaclav Kotesovec, May 22 2025 *)

PARI

a(n) = sum(i=0, 2*n, sum(j=0, 2*n-i, stirling(i+n, n, 2)*stirling(j+n, n,  2)*stirling(3*n-i-j, n, 2)));

Formula

a(n) = Sum_{i, j, k>=0 and i+j+k=2*n} Stirling2(i+n,n) * Stirling2(j+n,n) * Stirling2(k+n,n).

a(n) ~ 5^(5*n + 1) * n^(2*n - 1/2) / (sqrt(2*Pi*(1-w)) * 3^(3*n + 3/2) * exp(2*n) * (5-3*w)^(2*n) * w^(3*n+1)), where w = -LambertW(-5*exp(-5/3)/3) = 0.5404054440316473268672191331988... - Vaclav Kotesovec, May 22 2025