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A349256 G.f. A(x) satisfies A(x) = 1 / ((1 + x) * (1 - 3 * x * A(x)^2)).

%I A349256 #16 Aug 02 2023 09:40:05
%S A349256 1,2,19,206,2563,34415,486370,7128488,107364421,1651615568,
%T A349256 25840137724,409898503763,6577319627506,106571487893024,
%U A349256 1741193467526782,28653852176675324,474521786894159593,7902112425718228064,132243695376774536755,2222925664652778182060
%N A349256 G.f. A(x) satisfies A(x) = 1 / ((1 + x) * (1 - 3 * x * A(x)^2)).
%H A349256 Seiichi Manyama, <a href="/A349256/b349256.txt">Table of n, a(n) for n = 0..797</a>
%F A349256 a(n) = (-1)^n + 3 * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
%F A349256 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+k,n-k) * 3^k * binomial(3*k,k) / (2*k+1).
%F A349256 a(n) = (-1)^n*hypergeom([1/3, 2/3, -n, n + 1], [1/2, 1, 3/2], (3/2)^4). - _Peter Luschny_, Nov 12 2021
%F A349256 a(n) ~ sqrt(585 + 73*sqrt(65)) * (73 + 9*sqrt(65))^n / (3^(5/2) * sqrt(Pi) * n^(3/2) * 2^(3*n + 5/2)). - _Vaclav Kotesovec_, Nov 13 2021
%t A349256 nmax = 19; 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 A349256 a[n_] := a[n] = (-1)^n + 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, 19}]
%t A349256 Table[Sum[(-1)^(n - k) Binomial[n + k, n - k] 3^k Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 19}]
%t A349256 a[n_] := (-1)^n*HypergeometricPFQ[{1/3, 2/3, -n, n + 1}, {1/2, 1, 3/2}, (3/2)^4]; Table[a[n], {n, 0, 19}] (* _Peter Luschny_, Nov 12 2021 *)
%Y A349256 Cf. A001764, A107841, A217363, A337169, A346627, A349253, A349254, A349255.
%K A349256 nonn
%O A349256 0,2
%A A349256 _Ilya Gutkovskiy_, Nov 12 2021