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

%I #18 Mar 03 2024 09:50:30

%S 1,1,2,6,17,52,167,558,1912,6683,23736,85426,310861,1141837,4227938,

%T 15764474,59140089,223062670,845388258,3217750229,12295043520,

%U 47144444476,181349473833,699629022954,2706327445312,10494497061015,40787775234746,158859378070721

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

%F G.f.: A(x) = 2*(1 + x^3) / (1 + sqrt(1-4*x*(1 + x^3)^2)).

%F a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-6*k+1,k) * binomial(2*n-6*k+1,n-3*k) / (2*n-6*k+1).

%F D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +(n+1)*a(n-3) +6*(-2*n+7)*a(n-4) +6*(-2*n+13)*a(n-7) +2*(-2*n+19)*a(n-10)=0. - _R. J. Mathar_, Jul 25 2023

%p A364329 := proc(n)

%p add( binomial(2*n-6*k+1,k) * binomial(2*n-6*k+1,n-3*k)/(2*n-6*k+1),k=0..n/3) ;

%p end proc:

%p seq(A364329(n),n=0..70); # _R. J. Mathar_, Jul 25 2023

%t nmax = 27; A[_] = 1;

%t Do[A[x_] = (1 + x^3)*(1 + x*A[x]^2) + 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\3, binomial(2*n-6*k+1, k)*binomial(2*n-6*k+1, n-3*k)/(2*n-6*k+1));

%Y Cf. A073157, A215576, A364330.

%K nonn,easy

%O 0,3

%A _Seiichi Manyama_, Jul 18 2023