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A348957 G.f. A(x) satisfies A(x) = (1 + x * A(-x)) / (1 - x * A(x)).

%I A348957 #35 Aug 10 2025 07:08:11
%S A348957 1,2,2,10,18,98,210,1194,2786,16258,39906,236938,601458,3615330,
%T A348957 9399858,57024426,150947010,922283522,2475603138,15212318730,
%U A348957 41290579410,254909413218,698230131858,4327273358250,11943274468770,74260741616514,206279837823650,1286199407132554
%N A348957 G.f. A(x) satisfies A(x) = (1 + x * A(-x)) / (1 - x * A(x)).
%F A348957 a(0) = 1; a(n) = -(-1)^n * a(n-1) + Sum_{k=0..n-1} a(k) * a(n-k-1).
%F A348957 a(n) ~ c * 3^(3*n/4) * (1 + sqrt(3))^n / (sqrt(2*Pi) * n^(3/2) * 2^(n/2)), where c = 3^(1/4) if n is even and c = (1 + sqrt(3))/sqrt(2) if n is odd. - _Vaclav Kotesovec_, Nov 14 2021
%F A348957 From _Alexander Burstein_, Nov 26 2021: (Start)
%F A348957 G.f.: A(-x) = 1/A(x).
%F A348957 G.f.: A(x) = 1 + x*(1+A(x)^3)/A(x). (End)
%F A348957 a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(2*n-3*k-2,n-1) for n > 0. - _Seiichi Manyama_, Apr 11 2024
%t A348957 nmax = 27; A[_] = 0; Do[A[x_] = (1 + x A[-x])/(1 - x A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A348957 a[0] = 1; a[n_] := a[n] = -(-1)^n a[n - 1] + Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 27}]
%t A348957 CoefficientList[y/.AsymptoticSolve[y-y^2+x(1+y^3)==0,y->1,{x,0,27}][[1]],x] (* _Alexander Burstein_, Nov 26 2021 *)
%Y A348957 Cf. A006318, A086246.
%Y A348957 Cf. A349310, A361638, A364407, A366266, A366364.
%Y A348957 Cf. A112478, A364394, A364395, A366363.
%K A348957 nonn
%O A348957 0,2
%A A348957 _Ilya Gutkovskiy_, Nov 05 2021