Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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