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%I #2 Mar 30 2012 18:37:15
%S 1,2,9,48,278,1696,10736,69886,465019,3149476,21643433,150554144,
%T 1058101315,7502183626,53599160532,385494328218,2788827078507,
%U 20280590381098,148167425970522,1087007419753186,8004683588800899
%N G.f.: A(x) = F(x*G(x))^2 where F(x) = G(x*F(x)) = 1 + x*F(x)^3 is the g.f. of A001764 and G(x) = F(x/G(x)) = 1 + x*G(x)^2 is the g.f. of A000108 (Catalan).
%F a(n) = Sum_{k=0..n} C(3k+2,k)*2/(3k+2) * C(2n-k,n-k)*k/(2n-k) for n>0 with a(0)=1.
%F G.f. satisfies: A(x*F(x)) = F(x*F(x)^2)^2 where F(x) is the g.f. of A001764.
%e G.f.: A(x) = F(x*G(x))^2 = 1 + 2*x + 9*x^2 + 48*x^3 + 278*x^4 +... where
%e F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
%e F(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
%e F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 + 7752*x^6 +...
%e G(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
%e G(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
%o (PARI) {a(n)=if(n==0,1,sum(k=0,n,binomial(3*k+2,k)*2/(3*k+2)*binomial(2*(n-k)+k,n-k)*k/(2*(n-k)+k)))}
%Y Cf. A000108, A001764; A153299, A153391.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 15 2009
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