%I #9 Nov 03 2021 05:19:56
%S 1,1,2,6,29,221,2815,59607,2175115,134785987,14543011028,
%T 2682224473296,864129873439979,476879023670530355,
%U 460188677448639450646,761220053428592181980874,2202591080616789155249254723,10927081698418028875550581480027,94836180093445711611212497662570806
%N G.f. A(x) satisfies: A(x) = 1 / (1 - x - x^2 * A(3*x)).
%F a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-2} 3^k * a(k) * a(n-k-2).
%F a(n) ~ c * 3^(n*(n-2)/4), where c = 4.2101130581370834571021724998929772199905440992108887037121562184404379... - _Vaclav Kotesovec_, Nov 03 2021
%t nmax = 18; A[_] = 0; Do[A[x_] = 1/(1 - x - x^2 A[3 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[3^k a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 18}]
%Y Cf. A001006, A015084, A348878, A348880.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 02 2021