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%I #9 Nov 03 2021 04:44:56
%S 1,2,12,232,15792,4108192,4223439552,17316156716672,
%T 283777228606348032,18598759772257600748032,
%U 4875627680189345535622228992,5112485673116229482189477259405312,21443339558695300334256395183459423465472,359759625310995318218730673236935427042834358272
%N G.f. A(x) satisfies: A(x) = 1 / (1 - x - x * A(4*x)).
%F a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} 4^k * a(k) * a(n-k-1).
%F a(n) ~ c * 2^(n*(n-1)), where c = 2*Product_{j>=1} (4^j+1)/(4^j-1) = 3.938520707336538863894387393934531340132379924622409970534801850699757421... - _Vaclav Kotesovec_, Nov 03 2021
%t nmax = 13; A[_] = 0; Do[A[x_] = 1/(1 - x - x A[4 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[4^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 13}]
%Y Cf. A006318, A015085, A348875, A348876.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Nov 02 2021