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A168653 G.f. satisfies: A(x*A(x)) = G(x) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

%I A168653 #2 Mar 30 2012 18:37:20
%S A168653 1,1,2,6,21,82,340,1478,6622,30433,142331,676203,3248579,15776459,
%T A168653 77196573,380849394,1888606247,9430534212,47236684433,238214461960,
%U A168653 1202007809362,6116704517639,30997312336216,159384351652358
%N A168653 G.f. satisfies: A(x*A(x)) = G(x) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.
%F A168653 G.f. satisfies: A(x) = 1 + A(x)^3*Series_Reversion(x*A(x)).
%F A168653 G.f. satisfies: A( x*(1-x)^2*A(x*(1-x)^2) ) = 1/(1-x).
%F A168653 G.f. satisfies: A( (x/(1+x)^3)*A(x/(1+x)^3) ) = 1 + x.
%e A168653 G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 82*x^5 + 340*x^6 +...
%e A168653 A(x*A(x)) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
%o A168653 (PARI) {a(n)=local(A=1+x, F=sum(k=0, n, binomial(3*k+1, k)/(3*k+1)*x^k)+x*O(x^n)); for(i=0, n, A=subst(F, x, serreverse(x*(A+x*O(x^n))^1))); polcoeff(A, n)}
%o A168653 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+A^3*serreverse(x*(A+x*O(x^n)))); polcoeff(A,n)}
%Y A168653 Cf. A154677, A168478, A168448, A001764.
%K A168653 nonn
%O A168653 0,3
%A A168653 _Paul D. Hanna_, Dec 06 2009