%I #9 Nov 21 2021 05:25:27
%S 1,4,40,544,8512,144448,2584960,48026368,917535232,17911696384,
%T 355725727744,7164414312448,145983839272960,3003998986682368,
%U 62337412584669184,1303045468017786880,27411525832634269696,579884892273731436544,12328565505725394583552
%N G.f. A(x) satisfies: A(x) = (1 + 3 * x * A(x)^3) / (1 - x).
%F a(0) = 1; a(n) = a(n-1) + 3 * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
%F a(n) = Sum_{k=0..n} binomial(n+2*k,3*k) * binomial(3*k,k) * 3^k / (2*k+1).
%F a(n) ~ sqrt(13 + 7*3^(1/3) + 5*3^(2/3)) / (12 * sqrt(Pi) * n^(3/2) * (1 + 3^(4/3)/2 - 3^(5/3)/2)^n). - _Vaclav Kotesovec_, Nov 21 2021
%t nmax = 18; A[_] = 0; Do[A[x_] = (1 + 3 x A[x]^3)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t a[0] = 1; a[n_] := a[n] = a[n - 1] + 3 Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 18}]
%t Table[Sum[Binomial[n + 2 k, 3 k] Binomial[3 k, k] 3^k/(2 k + 1), {k, 0, n}], {n, 0, 18}]
%o (PARI) a(n) = sum(k=0, n, binomial(n+2*k,3*k) * binomial(3*k,k) * 3^k / (2*k+1)) \\ _Andrew Howroyd_, Nov 20 2021
%Y Cf. A001764, A103211, A346626, A348912, A349517.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Nov 20 2021