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%I #4 Jun 08 2021 07:19:48
%S 1,4,25,247,3283,54661,1092427,25473037,678837319,20351864821,
%T 677954261635,24842157250117,993040102321927,43003754679356941,
%U 2005536858420616963,100211634039201328381,5341144936822423446247,302468060262966258380773,18136282125753572653056355
%N a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1).
%F E.g.f. A(x) satisfies: A'(x) = 3 * A(x)^2 + exp(x).
%t a[n_] := a[n] = 1 + 3 Sum[Binomial[n - 1, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
%t nmax = 18; A[_] = 1; Do[A[x_] = Normal[Integrate[3 A[x]^2 + Exp[x], x] + O[x]^(nmax + 1)], nmax]; CoefficientList[A[x], x] Range[0, nmax]!
%Y Cf. A032031, A054687, A345102, A345104.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jun 08 2021