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 A384206 #8 May 22 2025 17:09:18
%S A384206 1,4,1291,2107596,9822847079,99559982844000,1870441451243408425,
%T A384206 58630795546429054116336,2846132741588198942785663319,
%U A384206 202389763024999232451527049522000,20194222519959431156536932169706390700,2731878423936456763814384150978735866605108
%N A384206 a(n) = [x^(3*n)] Product_{k=0..n} 1/(1 - k*x)^2.
%F A384206 a(n) = Sum_{k=0..3*n} Stirling2(n+k, n) * Stirling2(4*n-k, n).
%F A384206 a(n) ~ 5^(5*n + 1/2) * n^(3*n - 1/2) / (sqrt(Pi*(1-w)) * 2^(2*n + 3/2) * exp(3*n) * w^(2*n + 1/2) * (5 - 2*w)^(3*n)), where w = -LambertW(-5*exp(-5/2)/2) = 0.268388115976977211740078521072609338...
%t A384206 Table[SeriesCoefficient[Product[1/(1-k*x)^2, {k, 0, n}], {x, 0, 3*n}], {n, 0, 15}]
%t A384206 Table[Sum[StirlingS2[i+n, n] * StirlingS2[4*n-i, n], {i, 0, 3*n}], {n, 0, 15}]
%Y A384206 Cf. A350376, A384207.
%K A384206 nonn
%O A384206 0,2
%A A384206 _Vaclav Kotesovec_, May 22 2025