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%I #6 Nov 26 2021 05:08:14
%S 1,4,18,93,550,3636,26079,197931,1562382,12685116,105187512,886700898,
%T 7574331987,65413265014,570155069547,5008957733472,44306834969838,
%U 394269180748272,3527034255411864,31700659283908242,286124960854479888,2592334353741781752,23567790327842864046
%N G.f. A(x) satisfies: A(x) = 1 / (1 - 3*x) + x * (1 - 3*x) * A(x)^3.
%C Third binomial transform of A001764.
%F a(n) = Sum_{k=0..n} binomial(n,k) * binomial(3*k,k) * 3^(n-k) / (2*k + 1).
%F a(n) ~ 3^(n - 5/2) * 13^(n + 3/2) / (sqrt(Pi) * n^(3/2) * 2^(2*(n+1))). - _Vaclav Kotesovec_, Nov 26 2021
%t nmax = 22; A[_] = 0; Do[A[x_] = 1/(1 - 3 x) + x (1 - 3 x) A[x]^3 + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]
%t Table[Sum[Binomial[n, k] Binomial[3 k, k] 3^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 22}]
%t Table[3^n HypergeometricPFQ[{1/3, 2/3, -n}, {1, 3/2}, -9/4], {n, 0, 22}]
%Y Cf. A001764, A104455, A188687, A346762.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Aug 02 2021
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