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%I #10 Oct 05 2023 08:35:01
%S 1,2,6,25,110,520,2566,13073,68244,363129,1962304,10739914,59411546,
%T 331652408,1865903040,10569319231,60227702736,345015430415,
%U 1985747398748,11477353063881,66590427901454,387685469752989,2264180109124196,13261401158297918
%N G.f. A(x) satisfies A(x) = 1 + x + x*(1 + x)^2*A(x)^2.
%F G.f.: 2*(1+x) / (1 + sqrt(1-4*x*(1+x)^3)).
%F a(n) = Sum_{k=0..n} binomial(3*k+1,n-k) * binomial(2*k,k)/(k+1).
%o (PARI) a(n) = sum(k=0, n, binomial(3*k+1, n-k)*binomial(2*k, k)/(k+1));
%Y Cf. A073157, A366237, A366238.
%Y Cf. A000108.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Oct 05 2023
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