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 A383882 #9 May 23 2025 06:15:52
%S A383882 1,10,750,106470,22350954,6220194750,2157580085700,896587036640680,
%T A383882 434225240080346858,240175986308550372366,149377949042637543000150,
%U A383882 103192471874508023383125750,78394850841083734162487127720,64957213308036504429927388238088,58298851680969051596827194829579744
%N A383882 a(n) = [x^n] Product_{k=1..4*n} 1/(1 - k*x).
%C A383882 In general, for m>=1, Stirling2((m+1)*n, m*n) ~ (-1)^(m*n) * (m+1)^((m+1)*n) * n^(n - 1/2) / (sqrt(2*Pi*(1 + w(m))) * exp(n) * m^(m*n + 1/2) * w(m)^(m*n) * (1 + 1/m + w(m))^n), where w(m) = LambertW(-(1 + 1/m)/exp(1 + 1/m)).
%F A383882 a(n) = Stirling2(5*n,4*n).
%F A383882 a(n) ~ 5^(5*n) * n^(n - 1/2) / (sqrt(2*Pi*(1 + w)) * exp(n) * 4^(4*n + 1/2) * w^(4*n) * (5/4 + w)^n), where w = LambertW(-5/(4*exp(5/4))).
%t A383882 Table[SeriesCoefficient[Product[1/(1-k*x), {k, 1, 4*n}], {x, 0, n}], {n, 0, 15}]
%t A383882 Table[StirlingS2[5*n, 4*n], {n, 0, 15}]
%t A383882 Table[SeriesCoefficient[1/(Pochhammer[1 - 1/x, 4*n]*x^(4*n)), {x, 0, n}], {n, 0, 15}]
%Y A383882 Cf. A007820 (m=1), A348084 (m=2), A383881 (m=3).
%Y A383882 Cf. A217913.
%Y A383882 Cf. A187646, A384129, A384130.
%K A383882 nonn
%O A383882 0,2
%A A383882 _Vaclav Kotesovec_, May 13 2025