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%I #7 Nov 26 2021 04:29:37
%S 1,3,11,50,271,1655,10900,75388,539295,3954593,29557251,224308078,
%T 1723659436,13384272660,104855628776,827760536528,6578127170319,
%U 52581460222645,422478996770305,3410174204693310,27640220748529799,224866485110361767,1835589569664256976
%N G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x) * A(x)^3.
%C Second binomial transform of A001764.
%F a(n) = Sum_{k=0..n} binomial(n,k) * binomial(3*k,k) * 2^(n-k) / (2*k + 1).
%F a(n) ~ 35^(n + 3/2) / (81 * sqrt(Pi) * n^(3/2) * 4^(n+1)). - _Vaclav Kotesovec_, Nov 26 2021
%t nmax = 22; A[_] = 0; Do[A[x_] = 1/(1 - 2 x) + x (1 - 2 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] 2^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 22}]
%t Table[2^n HypergeometricPFQ[{1/3, 2/3, -n}, {1, 3/2}, -27/8], {n, 0, 22}]
%Y Cf. A001764, A064613, A188687, A346763.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Aug 02 2021
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