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 A384012 #22 May 19 2025 04:55:58
%S A384012 1,3,33,630,17247,616770,27264976,1436603616,87922855935,
%T A384012 6131105251425,479931312805425,41674568874964740,3975727750503656820,
%U A384012 413360925414308633034,46523118781014280909560,5635356193271621706436800,730994763063708819170060751,101099888222006502307905386445
%N A384012 a(n) = [x^n] Product_{k=0..n} (1 + k*x)^3.
%F A384012 a(n) = Sum_{0<=i, j, k<=n and i+j+k=2*n} |Stirling1(n+1,i+1) * Stirling1(n+1,j+1) * Stirling1(n+1,k+1)|.
%F A384012 a(n) ~ 3^(3*n + 3/2) * w^(3*n+2) * n^(n - 1/2) / (2^(2*n + 5/2) * sqrt(Pi*(w-1)) * exp(n) * (3*w-2)^n), where w = -LambertW(-1,-2*exp(-2/3)/3) = 1.4293552275170056487105688431034768889546376014196... - _Vaclav Kotesovec_, May 18 2025
%t A384012 Table[SeriesCoefficient[Product[(1+k*x)^3, {k, 1, n}], {x, 0, n}], {n, 0, 17}] (* _Vaclav Kotesovec_, May 18 2025 *)
%o A384012 (PARI) a(n) = sum(i=0, n, sum(j=0, 2*n-i, abs(stirling(n+1, i+1, 1)*stirling(n+1, j+1, 1)*stirling(n+1, 2*n-i-j+1, 1))));
%Y A384012 Cf. A129256, A384031, A351507, A384017, A383862.
%K A384012 nonn
%O A384012 0,2
%A A384012 _Seiichi Manyama_, May 17 2025