%I #13 Dec 04 2023 05:33:00

%S 1,1,2,6,17,52,168,561,1922,6719,23871,85938,312823,1149421,4257460,

%T 15880036,59594517,224856450,852491806,3245959002,12407332166,

%U 47592364107,183139542306,706794663136,2735053815771,10609811267757,41251228784198

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

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

%F a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k+1,k) * binomial(2*n-5*k,n-3*k)/(n-2*k+1).

%F D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +(-2*n+7)*a(n-3) +(n-8)*a(n-6)=0. - _R. J. Mathar_, Dec 04 2023

%p A367056 := proc(n)

%p add(binomial(n-2*k+1,k) * binomial(2*n-5*k,n-3*k)/(n-2*k+1),k=0..floor(n/3)) ;

%p end proc:

%p seq(A367056(n),n=0..70) ; # _R. J. Mathar_, Dec 04 2023

%o (PARI) a(n) = sum(k=0, n\3, binomial(n-2*k+1, k)*binomial(2*n-5*k, n-3*k)/(n-2*k+1));

%Y Cf. A049140, A071969, A125305, A216490, A367042.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 04 2023