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A153292 G.f.: A(x) = F(x*F(x)^2) where F(x) = 1 + x*F(x)^3 is the g.f. of A001764.

%I A153292 #2 Mar 30 2012 18:37:15
%S A153292 1,1,5,31,211,1516,11295,86423,675051,5361323,43171480,351709926,
%T A153292 2894115003,24022408477,200918146461,1691749323232,14329850844625,
%U A153292 122028162988698,1044131083377287,8972696721635997,77408293908402336
%N A153292 G.f.: A(x) = F(x*F(x)^2) where F(x) = 1 + x*F(x)^3 is the g.f. of A001764.
%F A153292 a(n) = Sum_{k=0..n} C(3k+1,k)/(3k+1) * C(3n-k,n-k)*2k/(3n-k) for n>0 with a(0)=1.
%F A153292 G.f. satisfies: A(x) = 1 + x*F(x)^2*A(x)^3 where F(x) is the g.f. of A001764.
%F A153292 G.f. satisfies: A(x/G(x)) = F(x*G(x)) where G(x) = F(x/G(x)) is the g.f. of A000108 and F(x) is the g.f. of A001764.
%e A153292 G.f.: A(x) = F(x*F(x)^2) = 1 + x + 5*x^2 + 31*x^3 + 211*x^4 +... where
%e A153292 F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
%e A153292 F(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
%e A153292 F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 + 7752*x^6 +...
%o A153292 (PARI) {a(n)=if(n==0,1,sum(k=0,n,binomial(3*k+1,k)/(3*k+1)*binomial(3*(n-k)+2*k,n-k)*2*k/(3*(n-k)+2*k)))}
%Y A153292 Cf. A001764, A000108; A153293.
%K A153292 nonn
%O A153292 0,3
%A A153292 _Paul D. Hanna_, Jan 14 2009