A384032 a(n) = [x^(2*n)] Product_{k=0..n} (1 + k*x)^4.
1, 6, 321, 46364, 13052881, 6077950570, 4237586784577, 4137911590389080, 5394217192300621089, 9055251708372687577550, 19032397641903957029149569, 48970167155426122072661229684, 151429299992138418402024853511537, 554184682895238619253412365302575346
Offset: 0
Keywords
Programs
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Mathematica
Table[Sum[StirlingS1[n+1, i+1] * StirlingS1[n+1, j+1] * StirlingS1[n+1, k+1] * StirlingS1[n+1, 2*n-i-j-k+1], {i, 0, n}, {j, 0, 2*n-i}, {k, 0, 2*n-i-j}], {n, 0, 15}] (* Vaclav Kotesovec, May 22 2025 *) -
PARI
a(n) = sum(i=0, n, sum(j=0, 2*n-i, sum(k=0, 2*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, 2*n-i-j-k+1, 1)))));
Formula
a(n) = Sum_{0<=i, j, k, l<=n and i+j+k+l=2*n} |Stirling1(n+1,i+1) * Stirling1(n+1,j+1) * Stirling1(n+1,k+1) * Stirling1(n+1,l+1)|.
a(n) ~ 2^(4*n + 1) * w^(4*n + 5/2) * n^(2*n - 1/2) / (sqrt(Pi*(w-1)) * exp(2*n) * (2*w-1)^(2*n)), where w = -LambertW(-1, -exp(-1/2)/2) = 1.7564312086261696769827376166... - Vaclav Kotesovec, May 22 2025