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 A382847 #11 Apr 08 2025 12:23:51
%S A382847 1,1,14,579,48044,6647405,1379024730,400315753159,154879704709784,
%T A382847 77018569697097009,47863427797633958630,36348262891572161261963,
%U A382847 33119479438137288670256964,35660343372397246917403353013,44791475616825872944740798413234,64911462519379469821754507087299215
%N A382847 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * (Stirling2(n,k) * k!)^2.
%F A382847 a(n) = (n!)^2 * [(x*y)^n] 1 / (exp(x) + exp(y) - exp(x + y))^n.
%F A382847 a(n) == 0 (mod n) for n > 0. - _Seiichi Manyama_, Apr 06 2025
%F A382847 a(n) ~ c * (r*(1+r)*(1 + 2*r + 2*sqrt(r*(1+r))))^n * n^(2*n) / exp(2*n), where r = 0.78386040488712123296193324113250946749673854534386788724235... is the root of the equation r = (1+r) * (1 + 1/(r*LambertW(-exp(-1/r)/r)))^2 and c = 0.947509273452712778524331973956110163137127694168427319... - _Vaclav Kotesovec_, Apr 08 2025
%t A382847 Table[Sum[Binomial[n + k - 1, k] (StirlingS2[n, k] k!)^2, {k, 0, n}], {n, 0, 15}]
%t A382847 Table[(n!)^2 SeriesCoefficient[1/(Exp[x] + Exp[y] - Exp[x + y])^n, {x, 0, n}, {y, 0, n}], {n, 0, 15}]
%o A382847 (PARI) a(n) = sum(k=0, n, binomial(n+k-1, k)*(k!*stirling(n, k, 2))^2); \\ _Seiichi Manyama_, Apr 06 2025
%Y A382847 Cf. A048144, A305919, A382737, A382738, A382739, A382853.
%K A382847 nonn
%O A382847 0,3
%A A382847 _Ilya Gutkovskiy_, Apr 06 2025