A384012 a(n) = [x^n] Product_{k=0..n} (1 + k*x)^3.
1, 3, 33, 630, 17247, 616770, 27264976, 1436603616, 87922855935, 6131105251425, 479931312805425, 41674568874964740, 3975727750503656820, 413360925414308633034, 46523118781014280909560, 5635356193271621706436800, 730994763063708819170060751, 101099888222006502307905386445
Offset: 0
Keywords
Programs
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Mathematica
Table[SeriesCoefficient[Product[(1+k*x)^3, {k, 1, n}], {x, 0, n}], {n, 0, 17}] (* Vaclav Kotesovec, May 18 2025 *) -
PARI
a(n) = sum(i=0, n, sum(j=0, 2*n-i, abs(stirling(n+1, i+1, 1)*stirling(n+1, j+1, 1)*stirling(n+1, 2*n-i-j+1, 1))));
Formula
a(n) = Sum_{0<=i, j, k<=n and i+j+k=2*n} |Stirling1(n+1,i+1) * Stirling1(n+1,j+1) * Stirling1(n+1,k+1)|.
a(n) ~ 3^(3*n + 3/2) * w^(3*n+2) * n^(n - 1/2) / (2^(2*n + 5/2) * sqrt(Pi*(w-1)) * exp(n) * (3*w-2)^n), where w = -LambertW(-1,-2*exp(-2/3)/3) = 1.4293552275170056487105688431034768889546376014196... - Vaclav Kotesovec, May 18 2025