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%I #17 Nov 03 2023 11:20:29
%S 1,1,2,6,16,48,152,500,1688,5816,20368,72288,259424,939808,3432192,
%T 12622416,46706144,173762016,649569216,2438748864,9191656192,
%U 34765298944,131912452864,501987944832,1915417307392,7326620001536,28088736525824,107913607531520
%N G.f. satisfies A(x) = 1 + x^3 + x*A(x)^2.
%F G.f.: A(x) = 2*(1+x^3) / (1+sqrt(1-4*x*(1+x^3))).
%F a(n) = Sum_{k=0..floor(n/3)} binomial(n-3*k+1,k) * binomial(2*(n-3*k),n-3*k)/(n-3*k+1).
%o (PARI) a(n) = sum(k=0, n\3, binomial(n-3*k+1, k)*binomial(2*(n-3*k), n-3*k)/(n-3*k+1));
%Y Cf. A366676, A367043.
%Y Cf. A025227, A176697.
%Y Cf. A226022.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Nov 03 2023
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