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%I #5 Jun 08 2021 07:19:41
%S 1,3,13,89,825,9601,134185,2188353,40788745,855303265,19927758377,
%T 510728051073,14279388168137,432505475357729,14107767947949289,
%U 493046896702987841,18380057918926012809,728005164671113691105,30531323352522247757225,1351567976217998536472833
%N a(n) = 1 + 2 * Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1).
%F E.g.f. A(x) satisfies: A'(x) = 2 * A(x)^2 + exp(x).
%t a[n_] := a[n] = 1 + 2 Sum[Binomial[n - 1, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 19}]
%t nmax = 19; A[_] = 1; Do[A[x_] = Normal[Integrate[2 A[x]^2 + Exp[x], x] + O[x]^(nmax + 1)], nmax]; CoefficientList[A[x], x] Range[0, nmax]!
%Y Cf. A000165, A052886, A054687, A345105.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jun 08 2021