A349254 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - 3 * x * A(x)^2)).

1, 4, 37, 478, 7159, 116497, 2000386, 35671756, 654218641, 12261271942, 233798163646, 4521194100541, 88458184054882, 1747850650032532, 34828329987024058, 699083528482636228, 14121906499195594537, 286877562430915732546, 5856866441794110926809

Offset: 0

Author

Ilya Gutkovskiy, Nov 12 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..746

Crossrefs

Cf. A001764, A103211, A199475, A217363, A337167, A349253, A349255, A349256.

Programs

Mathematica

nmax = 18; A[] = 0; Do[A[x] = 1/((1 - x) (1 - 3 x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
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}]
Table[Sum[Binomial[n + k, n - k] 3^k Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 18}]
a[n_] := HypergeometricPFQ[{1/3, 2/3, -n, n + 1}, {1/2, 1, 3/2}, -81/16];
Table[a[n], {n, 0, 18}] (* Peter Luschny, Nov 12 2021 *)

Formula

a(n) = 1 + 3 * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).

a(n) = Sum_{k=0..n} binomial(n+k,n-k) * 3^k * binomial(3*k,k) / (2*k+1).

a(n) = hypergeom([1/3, 2/3, -n, n + 1], [1/2, 1, 3/2], -(3/2)^4). - Peter Luschny, Nov 12 2021

a(n) ~ sqrt(873 + 89*sqrt(97)) * (89 + 9*sqrt(97))^n / (3^(5/2) * sqrt(Pi) * n^(3/2) * 2^(3*n + 5/2)). - Vaclav Kotesovec, Nov 13 2021