A384025 a(n) = [x^(2*n)] Product_{k=0..n} (1 + k*x)^3.
1, 3, 66, 3815, 424428, 77530530, 21106440064, 8021533034676, 4060456997959152, 2642189599046492000, 2149789283054191431744, 2139041823964877704864992, 2555760236856152336740829440, 3611539707805518014521602175296, 5958533262158042791156143146398464
Offset: 0
Keywords
Programs
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Mathematica
Table[Sum[StirlingS1[n+1, i+1] * StirlingS1[n+1, j+1] * StirlingS1[n+1, n-i-j+1], {i, 0, n}, {j, 0, n-i}], {n, 0, 15}] (* Vaclav Kotesovec, May 22 2025 *) -
PARI
a(n) = sum(i=0, n, sum(j=0, n-i, abs(stirling(n+1, i+1, 1)*stirling(n+1, j+1, 1)*stirling(n+1, n-i-j+1, 1))));
Formula
a(n) = Sum_{i, j, k>=0 and i+j+k=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^(2*n - 1/2) / (sqrt(2*Pi*(w-1)) * exp(2*n) * (3*w-1)^(2*n)), where w = -LambertW(-1, -exp(-1/3)/3) = 2.23714702777371681804347369... - Vaclav Kotesovec, May 22 2025