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 A371142 #15 Mar 12 2024 09:02:40
%S A371142 1,0,2,3,52,245,4446,40327,823432,11539593,269030530,5151458411,
%T A371142 137198342700,3328477618765,100753779655270,2948537330575215,
%U A371142 100684293424304656,3437897124964763921,131334298073705194506,5110990692709364143219,216634434142329948020020
%N A371142 E.g.f. satisfies A(x) = 1 + x*A(x)^2 * (exp(x) - 1).
%F A371142 E.g.f.: 2/(1 + sqrt(1-4*x*(exp(x)-1))).
%F A371142 a(n) = n! * Sum_{k=0..floor(n/2)} (2*k)!/(k+1)! * Stirling2(n-k,k)/(n-k)!.
%F A371142 a(n) ~ sqrt(2 + 2*r + 8*r^2) * n^(n-1) / (exp(n) * r^n), where r = 0.4454633431717774246390371283449007888959504623839... is the positive real root of the equation 4*r*(exp(r) - 1) = 1. - _Vaclav Kotesovec_, Mar 12 2024
%t A371142 nmax = 20; CoefficientList[Series[(1 - Sqrt[1 + 4*x - 4*E^x*x]) / (2*(-x + E^x*x)), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Mar 12 2024 *)
%o A371142 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(2/(1+sqrt(1-4*x*(exp(x)-1)))))
%o A371142 (PARI) a(n) = n!*sum(k=0, n\2, (2*k)!/(k+1)!*stirling(n-k, k, 2)/(n-k)!);
%Y A371142 Cf. A052848, A371143.
%Y A371142 Cf. A371140.
%K A371142 nonn
%O A371142 0,3
%A A371142 _Seiichi Manyama_, Mar 12 2024