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 A383282 #13 Apr 24 2025 04:37:56
%S A383282 1,1,5,51,906,24690,956790,49993650,3387124440,288755250840,
%T A383282 30247310482200,3818739956308200,571858101118458000,
%U A383282 100218359688123877200,20319306632495415745200,4719164981053010642154000,1244680987088062472732784000,369981708267221405777101680000
%N A383282 a(n) = Sum_{k=0..n} (2*k+1) * (-1/2)^(n+k) * (2*k)! * (n-k)! * binomial(n,k)^2.
%F A383282 a(n) = (-1)^n * (n!)^2 * Sum_{k=0..n} (1/2)^(n-k) * binomial(-3/2,k)/(n-k)!.
%F A383282 a(n) = (n!)^2 * [x^n] 1/(1-x)^(3/2) * exp(-x/2).
%F A383282 a(n) = n * ( n*a(n-1) + (n-1)^2/2 * a(n-2) ) for n > 1.
%F A383282 a(n) ~ 4 * sqrt(Pi) * n^(2*n + 3/2) / exp(2*n + 1/2). - _Vaclav Kotesovec_, Apr 24 2025
%o A383282 (PARI) a(n) = sum(k=0, n, (2*k+1)*(2*k)!*(n-k)!*binomial(n, k)^2/(-2)^(n+k));
%Y A383282 Cf. A383281.
%K A383282 nonn
%O A383282 0,3
%A A383282 _Seiichi Manyama_, Apr 22 2025