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%I #2 Mar 30 2012 18:37:15
%S 1,2,9,48,276,1656,10212,64190,409218,2637282,17143506,112224228,
%T 738926064,4889332266,32488240779,216664589058,1449568426292,
%U 9725637277248,65417353098837,441013558347228,2979206654245122
%N G.f.: A(x) = F(x*G(x)^2)^2 where F(x) = G(x/F(x)) = 1 + x*F(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(x*G(x)) = 1 + x*G(x)^3 is the g.f. of A001764.
%F a(n) = Sum_{k=0..n} C(2k+2,k)/(k+1) * C(3n-k,n-k)*2k/(3n-k) for n>0 with a(0)=1.
%F G.f. satisfies: A(x/F(x)) = F(x*F(x))^2 where F(x) is the g.f. of A000108 (Catalan).
%e G.f.: A(x) = F(x*G(x)^2)^2 = 1 + 2*x + 9*x^2 + 48*x^3 + 276*x^4 +... where
%e F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
%e F(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
%e G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
%e G(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
%e G(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 + 7752*x^6 +...
%o (PARI) {a(n)=if(n==0,1,sum(k=0,n,binomial(2*k+2,k)*2/(2*k+2)*binomial(3*(n-k)+2*k,n-k)*2*k/(3*(n-k)+2*k)))}
%Y Cf. A000108, A001764; A153296, A153298.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 15 2009
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