This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384032 #15 May 22 2025 13:20:38
%S A384032 1,6,321,46364,13052881,6077950570,4237586784577,4137911590389080,
%T A384032 5394217192300621089,9055251708372687577550,
%U A384032 19032397641903957029149569,48970167155426122072661229684,151429299992138418402024853511537,554184682895238619253412365302575346
%N A384032 a(n) = [x^(2*n)] Product_{k=0..n} (1 + k*x)^4.
%F A384032 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)|.
%F A384032 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
%t A384032 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 *)
%o A384032 (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)))));
%Y A384032 Cf. A382925, A384031.
%K A384032 nonn
%O A384032 0,2
%A A384032 _Seiichi Manyama_, May 17 2025