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%I #2 Mar 30 2012 18:37:15
%S 1,1,4,21,124,782,5145,34873,241682,1704240,12186900,88162753,
%T 644058237,4744733614,35210349041,262976828766,1975324849238,
%U 14913200362138,113107780322778,861417424802187,6585224638006020,50515048389265713
%N G.f.: A(x) = F(x*F(x)) where F(x) = 1 + x*F(x)^3 is the g.f. of A001764.
%F a(n) = Sum_{k=0..n} C(3k+1,k)/(3k+1) * C(3n-2k,n-k)*k/(3n-2k) for n>0 with a(0)=1.
%F G.f. satisfies: A(x) = 1 + x*F(x)*A(x)^3 where F(x) is the g.f. of A001764.
%F G.f. satisfies: A(x/G(x)) = F(x) where G(x) = F(x/G(x)) is the g.f. of A000108 and F(x) is the g.f. of A001764.
%e G.f.: A(x) = F(x*F(x)) = 1 + x + 4*x^2 + 21*x^3 + 124*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 +...
%o (PARI) {a(n)=if(n==0,1,sum(k=0,n,binomial(3*k+1,k)/(3*k+1)*binomial(3*(n-k)+k,n-k)*k/(3*(n-k)+k)))}
%Y Cf. A001764; A153292.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 14 2009
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