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 A383869 #15 May 17 2025 04:20:39
%S A383869 1,3,55,1890,95781,6427575,537306484,53791898160,6275077781973,
%T A383869 835898091070185,125195263380478655,20825548503275385870,
%U A383869 3809430011164368694260,759987002381075483922180,164221938436980055710082200,38209754165858724861944820000,9524153723280871205135022364485
%N A383869 a(n) = [x^n] 1/Product_{k=0..n} (1 - (n+k)*x).
%F A383869 a(n) = (1/n!) * Sum_{k=0..n} (-1)^(n-k) * (n+k)^(2*n) * binomial(n,k).
%F A383869 a(n) = Sum_{k=0..n} n^(n-k) * binomial(2*n,n+k) * Stirling2(n+k,n).
%F A383869 a(n) = Sum_{k=0..n} (-1)^k * (2*n)^(n-k) * binomial(2*n,n+k) * Stirling2(n+k,n).
%F A383869 a(n) ~ (r-1)^((r-1)*n) * (1+r)^(2*n + 1) * exp(n) * n^(n - 1/2) / (sqrt(2*Pi*(1 + (4-r)*r)) * r^(r*n)), where r = 2.106565648173949260853515992430777519716829316322... is the root of the equation exp(2/(1+r)) = r/(r-1). - _Vaclav Kotesovec_, May 17 2025
%t A383869 Table[SeriesCoefficient[1/Product[(1 - (n + k)*x), {k, 0, n}], {x, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, May 17 2025 *)
%o A383869 (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*(n+k)^(2*n)*binomial(n, k))/n!;
%Y A383869 Central terms of triangle A143395.
%Y A383869 Cf. A007820, A129506.
%K A383869 nonn
%O A383869 0,2
%A A383869 _Seiichi Manyama_, May 13 2025