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 A383862 #52 May 19 2025 04:56:26
%S A383862 1,3,48,1386,58278,3225915,221726711,18216234288,1741626159966,
%T A383862 189977753488050,23285057201978520,3168272346322892094,
%U A383862 473878954663846060735,77281168674525142984020,13647787698908399220563400,2594721838238358445753776000,528401900314147344955336365822
%N A383862 a(n) = [x^n] Product_{k=0..n} 1/(1 - k*x)^3.
%F A383862 a(n) = Sum_{i, j, k>=0 and i+j+k=n} Stirling2(i+n,n) * Stirling2(j+n,n) * Stirling2(k+n,n).
%F A383862 a(n) ~ 2^(8*n + 3/2) * n^(n - 1/2) / (sqrt(Pi*(1-w)) * exp(n) * 3^(3*n + 3/2) * (4 - 3*w)^n * w^(3*n + 1)), where w = -LambertW(-4*exp(-4/3)/3) = 0.727473355414332993149219573314579663... - _Vaclav Kotesovec_, May 18 2025
%t A383862 Table[SeriesCoefficient[Product[1/(1-k*x)^3, {k, 1, n}], {x, 0, n}], {n, 0, 16}] (* _Vaclav Kotesovec_, May 18 2025 *)
%t A383862 (* or *)
%t A383862 Table[Sum[StirlingS2[i + n, n] * StirlingS2[j + n, n] * StirlingS2[2*n - i - j, n], {i, 0, n}, {j, 0, n-i}], {n, 0, 16}] (* _Vaclav Kotesovec_, May 18 2025 *)
%o A383862 (PARI) a(n) = sum(i=0, n, sum(j=0, n-i, stirling(i+n, n, 2)*stirling(j+n, n, 2)*stirling(2*n-i-j, n, 2)));
%Y A383862 Cf. A007820, A350376, A384022, A384060.
%Y A383862 Cf. A384012.
%K A383862 nonn
%O A383862 0,2
%A A383862 _Seiichi Manyama_, May 17 2025