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 A384031 #19 May 19 2025 04:54:48
%S A384031 1,4,62,1680,65446,3334800,210218956,15803243456,1380404187558,
%T A384031 137419388080920,15359405910256580,1904647527097204032,
%U A384031 259511601503239509004,38539384808775589973416,6195988524478342471690200,1072149116496356641327200000,198683315255720972000976370950
%N A384031 a(n) = [x^n] Product_{k=0..n} (1 + k*x)^4.
%F A384031 a(n) = Sum_{0<=i, j, k, l<=n and i+j+k+l=3*n} |Stirling1(n+1,i+1) * Stirling1(n+1,j+1) * Stirling1(n+1,k+1) * Stirling1(n+1,l+1)|.
%F A384031 a(n) ~ 2^(8*n + 7/2) * w^(4*n + 5/2) * n^(n - 1/2) / (sqrt(Pi*(w-1)) * 3^(3*n + 5/2) * exp(n) * (4*w-3)^n), where w = -LambertW(-1,-3*exp(-3/4)/4) = 1.300200741659068588153265179374583756429... - _Vaclav Kotesovec_, May 18 2025
%t A384031 Table[SeriesCoefficient[Product[(1+k*x)^4, {k, 1, n}], {x, 0, n}], {n, 0, 16}] (* _Vaclav Kotesovec_, May 18 2025 *)
%o A384031 (PARI) a(n) = sum(i=0, n, sum(j=0, 3*n-i, sum(k=0, 3*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, 3*n-i-j-k+1, 1)))));
%Y A384031 Cf. A129256, A384012, A384017.
%Y A384031 Cf. A384060.
%Y A384031 Cf. A382925, A384032, A384029, A351507.
%K A384031 nonn
%O A384031 0,2
%A A384031 _Seiichi Manyama_, May 17 2025