%I #9 Jul 30 2023 09:56:39
%S 1,3,7,16,38,95,249,678,1901,5451,15906,47066,140868,425657,1296665,
%T 3977684,12276617,38094013,118768915,371875752,1168843808,3686549845,
%U 11664123048,37011249678,117750111763,375529083267,1200327617200,3844662925222,12338289374046
%N G.f. satisfies A(x) = 1/(1-x)^3 + x^2*A(x)^2.
%F G.f.: A(x) = 2 / ( (1-x)^3 * (1 + sqrt( 1 - 4*x^2/(1-x)^3 )) ).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n+k+2,3*k+2) * binomial(2*k,k) / (k+1).
%o (PARI) my(N=30, x='x+O('x^N)); Vec(2/((1-x)^3*(1+sqrt(1-4*x^2/(1-x)^3))))
%Y Cf. A000108, A086615, A162481, A360045.
%Y Cf. A364626, A364627.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Jul 30 2023