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 A341196 #19 Feb 20 2021 09:00:49
%S A341196 0,1,20,243,2800,33425,424116,5762155,83891648,1306561185,21709011700,
%T A341196 383654149571,7189789929840,142465285362673,2976697773182420,
%U A341196 65417312929686075,1508567496105346816,36425941390897897025,919100609186531702868
%N A341196 a(n) = Sum_{k=0..n} k^4 * (n-k)! * binomial(n,k)^2.
%C A341196 In general, for m >= 0, Sum_{k=0..n} k^m * (n-k)! * binomial(n,k)^2 ~ exp(2*sqrt(n) - n - 1/2) * n^(n + (2*m+1)/4) / sqrt(2). - _Vaclav Kotesovec_, Feb 20 2021
%F A341196 E.g.f.: x*(1+4*x-5*x^2+x^3)*exp(x/(1-x))/(1-x)^5.
%F A341196 a(n) = n*n!*F([2, 2, 1 - n], [1, 1, 1], -1), where F is the generalized hypergeometric function. - _Stefano Spezia_, Feb 06 2021
%F A341196 a(n) ~ exp(2*sqrt(n) - n - 1/2) * n^(n + 9/4) / sqrt(2). - _Vaclav Kotesovec_, Feb 20 2021
%t A341196 a[n_] := Sum[k^4 * (n-k)! * Binomial[n, k]^2, {k, 0, n}]; Array[a, 19, 0] (* _Amiram Eldar_, Feb 06 2021 *)
%o A341196 (PARI) a(n) = sum(k=0, n, k^4*(n-k)!*binomial(n, k)^2);
%o A341196 (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(x*(1+4*x-5*x^2+x^3)*exp(x/(1-x))/(1-x)^5)))
%Y A341196 Column 4 of A341200.
%K A341196 nonn
%O A341196 0,3
%A A341196 _Seiichi Manyama_, Feb 06 2021