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A153392 G.f.: A(x) = F(x*G(x)^3) 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).

%I A153392 #2 Mar 30 2012 18:37:15
%S A153392 1,1,6,39,272,2001,15333,121266,983274,8133564,68382628,582700485,
%T A153392 5021538753,43690059657,383263396836,3386175566418,30104702903914,
%U A153392 269125162789764,2417709649413102,21815252320257250,197620659225838530
%N A153392 G.f.: A(x) = F(x*G(x)^3) 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 A153392 a(n) = Sum_{k=0..n} C(3k+1,k)/(3k+1) * C(2n+k,n-k)*3k/(2n+k) for n>0 with a(0)=1.
%F A153392 G.f. satisfies: A(x) = 1 + x*G(x)^3*A(x)^3 where G(x) is the g.f. of A000108.
%F A153392 G.f. satisfies: A(x*F(x)) = F(x*F(x)^4) where F(x) is the g.f. of A001764.
%e A153392 G.f.: A(x) = F(x*G(x)^3) = 1 + x + 6*x^2 + 39*x^3 + 272*x^4+... where
%e A153392 F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
%e A153392 F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 +...
%e A153392 G(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
%e A153392 G(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
%e A153392 G(x)^3 = 1 + 3*x + 9*x^2 + 28*x^3 + 90*x^4 + 297*x^5 + 1001*x^6 +...
%e A153392 A(x)^2 = 1 + 2*x + 13*x^2 + 90*x^3 + 658*x^4 + 5014*x^5 +...
%e A153392 A(x)^3 = 1 + 3*x + 21*x^2 + 154*x^3 + 1176*x^4 + 9264*x^5 +...
%e A153392 G(x)^3*A(x)^3 = 1 + 6*x + 39*x^2 + 272*x^3 + 2001*x^4 + 15333*x^5 +...
%o A153392 (PARI) {a(n)=if(n==0,1,sum(k=0,n,binomial(3*k+1,k)/(3*k+1)*binomial(2*(n-k)+3*k,n-k)*3*k/(2*(n-k)+3*k)))}
%Y A153392 Cf. A000108, A001764; A153391, A153393.
%K A153392 nonn
%O A153392 0,3
%A A153392 _Paul D. Hanna_, Jan 15 2009