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%I #2 Mar 30 2012 18:37:15
%S 1,1,5,33,245,1941,16023,136075,1179833,10392981,92701411,835271032,
%T 7589337123,69444928453,639280878401,5915683250220,54991636090761,
%U 513257729193329,4807619948647095,45177320023095160,425766248463523359
%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)^4 is the g.f. of A002293.
%F a(n) = Sum_{k=0..n} C(3k+1,k)/(3k+1) * C(4n-2k,n-k)*k/(2n-k) for n>0 with a(0)=1.
%F G.f. satisfies: A(x) = 1 + x*G(x)^2*A(x)^3 where G(x) is the g.f. of A002293.
%F G.f. satisfies: A(x/F(x)) = F(x*F(x)) where F(x) is the g.f. of A001764.
%F G.f. satisfies: A(x/H(x)^2) = F(x) where H(x) = 1 + x*H(x)^2 is the g.f. of A000108 and F(x) is the g.f. of A001764.
%e G.f.: A(x) = F(x*G(x)^2) = 1 + x + 5*x^2 + 33*x^3 + 245*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 + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
%e G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 +...
%e G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 +...
%e G(x)^4 = 1 + 4*x + 22*x^2 + 140*x^3 + 969*x^4 + 7084*x^5 +...
%e A(x)^2 = 1 + 2*x + 11*x^2 + 76*x^3 + 581*x^4 + 4702*x^5 +...
%e A(x)^3 = 1 + 3*x + 18*x^2 + 130*x^3 + 1023*x^4 + 8457*x^5 +...
%e G(x)^2*A(x)^3 = 1 + 5*x + 33*x^2 + 245*x^3 + 1941*x^4 + 16023*x^5 +...
%o (PARI) {a(n)=if(n==0,1,sum(k=0,n,binomial(3*k+1,k)/(3*k+1)*binomial(4*(n-k)+2*k,n-k)*2*k/(4*(n-k)+2*k)))}
%Y Cf. A000108, A001764, A002293; A153397, A153399.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 15 2009
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