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%I #10 Nov 18 2024 09:44:43
%S 1,0,0,1,3,3,3,12,30,45,75,192,436,798,1554,3542,7740,15543,32183,
%T 70794,153252,321431,684123,1491504,3232672,6928779,14957787,32615388,
%U 70991040,153985890,335256886,733206840,1603258134,3503385568,7671749664,16837946850
%N G.f. A(x) satisfies A(x) = 1 + (x * (1+x))^3 * A(x)^2.
%F a(n) = Sum_{k=0..floor(n/3)} binomial(3*k,n-3*k) * C(k), where C(k) are the Catalan numbers (A000108).
%F G.f.: 2/(1 + sqrt(1 - 4*(x*(1+x))^3)).
%o (PARI) a(n) = sum(k=0, n\3, binomial(3*k, n-3*k)*binomial(2*k, k)/(k+1));
%Y Cf. A115055, A378151.
%Y Cf. A000108.
%K nonn
%O 0,5
%A _Seiichi Manyama_, Nov 18 2024