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 A363744 #16 Nov 18 2023 05:00:53
%S A363744 1,1,5,31,313,3981,63841,1223419,27378737,701091001,20221662241,
%T A363744 649032795951,22945630163017,886151307346501,37121193546044609,
%U A363744 1676607954371120611,81222976991097364321,4201418329450141471473,231127287514383805458625
%N A363744 E.g.f. satisfies A(x) = exp(x * (1 + x * A(x))^2).
%F A363744 a(n) = n! * Sum_{k=0..n} (n-k+1)^(k-1) * binomial(2*k,n-k)/k!.
%F A363744 a(n) ~ sqrt((1 + r*s)*(1 + 3*r*s) / (2*(1 + 2*r + 4*r^2*s + 2*r^3*s^2))) * n^(n-1) / (exp(n) * r^(n+1)), where r = 0.302307732979052080722256232095444259577495... and s = 2.910394288602135748195482733301939282588478379746... are real roots of the system of equations exp(r*(1 + r*s)^2) = s, 2*s*r^2*(1 + r*s) = 1. - _Vaclav Kotesovec_, Nov 18 2023
%t A363744 Join[{1}, Table[n! * Sum[(n-k+1)^(k-1) * Binomial[2*k,n-k]/k!, {k,0,n}], {n,1,20}]] (* _Vaclav Kotesovec_, Nov 18 2023 *)
%o A363744 (PARI) a(n) = n!*sum(k=0, n, (n-k+1)^(k-1)*binomial(2*k, n-k)/k!);
%Y A363744 Cf. A125500, A365030.
%Y A363744 Cf. A161635.
%K A363744 nonn
%O A363744 0,3
%A A363744 _Seiichi Manyama_, Aug 17 2023