A384207 a(n) = [x^(3*n)] Product_{k=0..n} 1/(1 - k*x)^3.

1, 10, 6562, 21157758, 192817813260, 3803916720008250, 138757892706447212551, 8432782489668636227456524, 792912489591430219972681508172, 109146372957847294924041235504625400, 21071987342698034891951000233099719150440, 5513873439400596105839885628799257242723984298

Offset: 0

Author

Vaclav Kotesovec, May 22 2025

Keywords

• nonn

Crossrefs

Cf. A007820, A383862, A384022, A384206.

Programs

Mathematica

Table[SeriesCoefficient[Product[1/(1-k*x)^3, {k, 0, n}], {x, 0, 3*n}], {n, 0, 15}]
Table[Sum[StirlingS2[i+n, n] * StirlingS2[j+n, n] * StirlingS2[4*n-i-j, n], {i, 0, 3*n}, {j, 0, 3*n-i}], {n, 0, 15}]

Formula

a(n) = Sum_{i=0..3*n, j=0..3*n-i} Stirling2(i+n, n) * Stirling2(j+n, n) * Stirling2(4*n-i-j, n).

a(n) ~ 2^(6*n + 1/2) * n^(3*n - 1/2) / (sqrt(3*Pi*(1-w)) * w^(3*n+1) * exp(3*n) * (2-w)^(3*n)), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.406375739959959907676958124124839758210...