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

%I A349254 #15 Aug 01 2023 09:33:49
%S A349254 1,4,37,478,7159,116497,2000386,35671756,654218641,12261271942,
%T A349254 233798163646,4521194100541,88458184054882,1747850650032532,
%U A349254 34828329987024058,699083528482636228,14121906499195594537,286877562430915732546,5856866441794110926809
%N A349254 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - 3 * x * A(x)^2)).
%H A349254 Seiichi Manyama, <a href="/A349254/b349254.txt">Table of n, a(n) for n = 0..746</a>
%F A349254 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 A349254 a(n) = Sum_{k=0..n} binomial(n+k,n-k) * 3^k * binomial(3*k,k) / (2*k+1).
%F A349254 a(n) = hypergeom([1/3, 2/3, -n, n + 1], [1/2, 1, 3/2], -(3/2)^4). - _Peter Luschny_, Nov 12 2021
%F A349254 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
%t A349254 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]
%t A349254 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 A349254 Table[Sum[Binomial[n + k, n - k] 3^k Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 18}]
%t A349254 a[n_] := HypergeometricPFQ[{1/3, 2/3, -n, n + 1}, {1/2, 1, 3/2}, -81/16];
%t A349254 Table[a[n], {n, 0, 18}] (* _Peter Luschny_, Nov 12 2021 *)
%Y A349254 Cf. A001764, A103211, A199475, A217363, A337167, A349253, A349255, A349256.
%K A349254 nonn
%O A349254 0,2
%A A349254 _Ilya Gutkovskiy_, Nov 12 2021