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 A332776 #29 Jun 11 2020 07:39:04
%S A332776 1,1,2,5,18,83,464,3041,22810,192595,1807328,18658097,210138882,
%T A332776 2563990859,33691089824,474327797585,7123141539610,113656386574099,
%U A332776 1920170741071280,34242622099969217,642792206343361602,12669617513914228907,261613287097165614224,5647565141926833774977
%N A332776 a(n) = 1 + Sum_{k=1..n-1} binomial(n-1,k) * a(k) * a(n-k-1).
%H A332776 Seiichi Manyama, <a href="/A332776/b332776.txt">Table of n, a(n) for n = 0..454</a>
%F A332776 E.g.f. A(x) satisfies: d/dx A(x) = exp(x) + A(x) * (A(x) - 1).
%F A332776 From _Vaclav Kotesovec_, Jun 09 2020: (Start)
%F A332776 E.g.f.: exp(x/2) * (BesselJ(2, 2*exp(x/2)) * BesselY(0,2) - BesselJ(0,2) * BesselY(2, 2*exp(x/2))) / (BesselJ(1, 2*exp(x/2)) * BesselY(0,2) - BesselJ(0,2) * BesselY(1, 2*exp(x/2))).
%F A332776 a(n) ~ n! / r^(n+1), where r = 1.0654335847261788612657252860730850911833168584... is the smallest real root of the equation BesselJ(1, 2*exp(r/2)) * BesselY(0,2) = BesselJ(0,2) * BesselY(1, 2*exp(r/2)). (End)
%t A332776 a[n_] := a[n] = 1 + Sum[Binomial[n - 1, k] a[k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]
%t A332776 terms = 23; A[_] = 0; Do[A[x_] = Normal[Integrate[Exp[x] + A[x] (A[x] - 1), x] + O[x]^(terms + 1)], terms]; CoefficientList[A[x], x] Range[0, terms]!
%Y A332776 Cf. A000110, A006677, A007317, A080635, A105633, A186021.
%K A332776 nonn
%O A332776 0,3
%A A332776 _Ilya Gutkovskiy_, Jun 08 2020