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A153394 G.f.: A(x) = F(x*G(x)^2)^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 A153394 #2 Mar 30 2012 18:37:15
%S A153394 1,3,18,118,813,5799,42470,317637,2416671,18649874,145655292,
%T A153394 1149199212,9146686605,73354982763,592217363334,4809250320023,
%U A153394 39258457746069,321964620209940,2651536017682988,21919266484180533,181820251665093357
%N A153394 G.f.: A(x) = F(x*G(x)^2)^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 A153394 a(n) = Sum_{k=0..n} C(3k+3,k)/(k+1) * C(2n,n-k)*k/n for n>0 with a(0)=1.
%F A153394 G.f. satisfies: A(x*F(x)) = F(x*F(x)^3)^3 = F(F(x)-1)^3 where F(x) is the g.f. of A001764.
%e A153394 G.f.: A(x) = F(x*G(x)^2)^3 = 1 + 3*x + 18*x^2 + 118*x^3 + 813*x^4 +... where
%e A153394 F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
%e A153394 F(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
%e A153394 F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 +...
%e A153394 G(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
%e A153394 G(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
%o A153394 (PARI) {a(n)=if(n==0,1,sum(k=0,n,binomial(3*k+3,k)/(k+1)*binomial(2*n,n-k)*k/n))}
%Y A153394 Cf. A000108, A001764; A153393, A153395.
%K A153394 nonn
%O A153394 0,2
%A A153394 _Paul D. Hanna_, Jan 15 2009