%I #9 Nov 21 2021 04:44:00
%S 1,4,24,192,1792,18240,196224,2194176,25247232,296979456,3555010560,
%T 43165900800,530362220544,6581594275840,82373440339968,
%U 1038579580796928,13179023462498304,168183976239562752,2157085003249876992,27790652486543474688,359485965093121818624
%N G.f. A(x) satisfies: A(x) = (1 + x * A(x)^3) / (1 - 3 * x).
%F a(0) = 1; a(n) = 3 * a(n-1) + 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^(n-k) / (2*k+1).
%F a(n) ~ (3/4*(7 + (3*(69 + 16*sqrt(3)))^(1/3) + (3*(69 - 16*sqrt(3)))^(1/3)))^n / (sqrt((4 - (2 + sqrt(3))^(1/3) - (2 - sqrt(3))^(1/3)) * Pi) * n^(3/2)). - _Vaclav Kotesovec_, Nov 21 2021
%t nmax = 20; A[_] = 0; Do[A[x_] = (1 + x A[x]^3)/(1 - 3 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t a[0] = 1; a[n_] := a[n] = 3 a[n - 1] + 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, 20}]
%t Table[Sum[Binomial[n + 2 k, 3 k] Binomial[3 k, k] 3^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 20}]
%o (PARI) a(n) = sum(k=0, n, binomial(n+2*k,3*k) * binomial(3*k,k) * 3^(n-k) / (2*k+1)) \\ _Andrew Howroyd_, Nov 20 2021
%Y Cf. A001764, A082298, A346626, A348793, A349515.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Nov 20 2021