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 A309473 #78 Jun 13 2020 00:52:54
%S A309473 1,0,1,1,3,11,43,195,1063,6395,42371,311883,2501159,21672355,
%T A309473 202544323,2028522067,21658255431,245738583307,2952508103651,
%U A309473 37440976938875,499785548010759,7005210659040979,102862231664567651,1579045889274408259,25294106622048460903
%N A309473 a(n) = (-1)^n + Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1).
%F A309473 E.g.f. A(x) satisfies: A'(x) = A(x)^2 - exp(-x).
%F A309473 From _Vaclav Kotesovec_, Jun 11 2020: (Start)
%F A309473 E.g.f.: exp(-x/2) * (BesselI(1, 2*exp(-x/2)) * (BesselK(0, 2) + BesselK(1, 2)) + (BesselI(0, 2) - BesselI(1, 2)) * BesselK(1, 2*exp(-x/2))) / ((BesselI(1, 2) - BesselI(0, 2)) * BesselK(0, 2*exp(-x/2)) + BesselI(0, 2*exp(-x/2)) * (BesselK(0, 2) + BesselK(1, 2))).
%F A309473 a(n) ~ n! / r^(n+1), where r = 1.4982609322383959128764444062824740935658895762... is the real root of the equation (BesselI(0, 2) - BesselI(1, 2)) * BesselK(0, 2*exp(-r/2)) = (BesselK(0, 2) + BesselK(1, 2)) * BesselI(0, 2*exp(-r/2)). (End)
%t A309473 a[n_] := a[n] = (-1)^n + Sum[Binomial[n - 1, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 24}]
%t A309473 terms = 24; A[_] = 1; Do[A[x_] = Normal[Integrate[A[x]^2 - Exp[-x], x] + O[x]^(terms + 1)], terms]; CoefficientList[A[x], x] Range[0, terms]!
%Y A309473 Cf. A006677, A014304, A054687, A332776, A335441.
%K A309473 nonn
%O A309473 0,5
%A A309473 _Ilya Gutkovskiy_, Jun 11 2020