A382925 a(n) = [x^(3*n)] Product_{k=0..n} (1 + k*x)^4.
1, 4, 248, 61320, 39194896, 51699564000, 122482878310656, 474300956527856640, 2804126507444905046272, 24036712401508315774848000, 286889291626307627568309995520, 4615084616716397442547883972818944, 97421519516367186622078306709619806208
Offset: 0
Keywords
Programs
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Mathematica
Abs[Table[Sum[StirlingS1[n+1, i+1] * StirlingS1[n+1, j+1] * StirlingS1[n+1, k+1] * StirlingS1[n+1, n-i-j-k+1], {i, 0, n}, {j, 0, n-i}, {k, 0, n-i-j}], {n, 0, 15}]] (* Vaclav Kotesovec, May 22 2025 *) -
PARI
a(n) = sum(i=0, n, sum(j=0, n-i, sum(k=0, n-i-j, abs(stirling(n+1, i+1, 1)*stirling(n+1, j+1, 1)*stirling(n+1, k+1, 1)*stirling(n+1, n-i-j-k+1, 1)))));
Formula
a(n) = Sum_{i, j, k, l>=0 and i+j+k+l=n} |Stirling1(n+1,i+1) * Stirling1(n+1,j+1) * Stirling1(n+1,k+1) * Stirling1(n+1,l+1)|.
a(n) ~ 2^(8*n + 7/2) * w^(4*n + 5/2) * n^(3*n - 1/2) / (sqrt(Pi*(w-1)) * exp(3*n) * (4*w-1)^(3*n)), where w = -LambertW(-1, -exp(-1/4)/4) = 2.5866629822630538811828... - Vaclav Kotesovec, May 22 2025