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G.f. A(x) satisfies A(x) = 1 + x*(1 + x)^2*A(x)^3.
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%I #17 Mar 03 2024 14:54:16

%S 1,1,5,25,145,905,5941,40433,282721,2018897,14661349,107945993,

%T 803922289,6045458905,45840518933,350100674785,2690717983169,

%U 20794719218593,161502488175557,1259855507859193,9867012143508305,77554946281194793,611575725258403061

%N G.f. A(x) satisfies A(x) = 1 + x*(1 + x)^2*A(x)^3.

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

%F G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366434.

%t nmax = 22; A[_] = 1;

%t Do[A[x_] = 1 + x*(1 + x)^2*A[x]^3 + O[x]^(nmax+1) // Normal, {nmax+1}];

%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Mar 03 2024 *)

%o (PARI) a(n) = sum(k=0, n, binomial(2*k, n-k)*binomial(3*k, k)/(2*k+1));

%Y Cf. A364475, A366200, A366222.

%Y Cf. A002478, A073155.

%Y Cf. A366176, A366434.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Oct 04 2023