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 A384205 #10 May 22 2025 17:07:03
%S A384205 1,3,201,40792,16904053,11861321255,12632193171300,19003969060842360,
%T A384205 38387884967440214085,100260769162534336491025,
%U A384205 328834941448280603509191681,1323249839691864496146379353852,6410573322270839015074278503521740,36805304509116365389123823470306765972
%N A384205 a(n) = [x^(2*n)] Product_{k=0..n} 1/(1 - k*x)^2.
%F A384205 a(n) = Sum_{k=0..2*n} Stirling2(n+k, n) * Stirling2(3*n-k, n).
%F A384205 a(n) ~ 2^(4*n - 1/2) * n^(2*n - 1/2) / (sqrt(Pi*(1-w)) * exp(2*n) * (2-w)^(2*n) * w^(2*n + 1/2)), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599...
%t A384205 Table[SeriesCoefficient[Product[1/(1-k*x)^2, {k, 0, n}], {x, 0, 2*n}], {n, 0, 15}]
%t A384205 Table[Sum[StirlingS2[i+n, n] * StirlingS2[3*n-i, n], {i, 0, 2*n}], {n, 0, 15}]
%Y A384205 Cf. A007820, A350376, A384206.
%K A384205 nonn
%O A384205 0,2
%A A384205 _Vaclav Kotesovec_, May 22 2025