A349531 - OEIS

%I #9 Nov 25 2021 03:02:03

%S 1,1,6,39,271,1986,15171,119694,968589,7997970,67132164,571138362,

%T 4914229293,42690269053,373915274505,3298492524831,29279961769422,

%U 261348675838758,2344226713167048,21119556517672650,191022983002755162,1733959471178342088,15790787266617518790

%N G.f. A(x) satisfies: A(x) = 1 + x * A(x)^3 / (1 - 3 * x).

%F a(0) = a(1) = 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-1,k-1) * binomial(3*k,k) * 3^(n-k) / (2*k+1).

%F a(n) = 3^(n-1)*F([4/3, 5/3, 1-n], [2, 5/2], -(3/2)^2), where F is the generalized hypergeometric function. - _Stefano Spezia_, Nov 21 2021

%F a(n) ~ 3^(n - 1/2) * 13^(n + 1/2) / (sqrt(Pi) * n^(3/2) * 2^(2*n + 2)). - _Vaclav Kotesovec_, Nov 25 2021

%t nmax = 22; 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] = a[1] = 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, 22}]

%t Table[Sum[Binomial[n - 1, k - 1] Binomial[3 k, k] 3^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 22}]

%Y Cf. A001764, A182401, A307678, A349318, A349532.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Nov 21 2021