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A168594 G.f. A(x) satisfies: A(x) = F(x/A(x)) where A(x*F(x)) = F(x) = g.f. of A133053, which is the squares of Motzkin numbers (A001006).

%I A168594 #2 Mar 30 2012 18:37:20
%S A168594 1,1,3,6,20,70,302,1386,6902,35862,194202,1082642,6191680,36141118,
%T A168594 214715244,1294849186,7911159522,48888093910,305165808290,
%U A168594 1921992409066,12202404037088,78031629139246,502263432618224,3252160882871210
%N A168594 G.f. A(x) satisfies: A(x) = F(x/A(x)) where A(x*F(x)) = F(x) = g.f. of A133053, which is the squares of Motzkin numbers (A001006).
%F A168594 G.f.: A(x) = x/Series_Reversion(x*F(x)) where F(x) = g.f. of A133053.
%e A168594 G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 20*x^4 + 70*x^5 + 302*x^6 +...
%e A168594 A(x) satisfies: A(x*F(x)) = F(x) = g.f. of A133053:
%e A168594 F(x) = 1 + x + 4*x^2 + 16*x^3 + 81*x^4 + 441*x^5 + 2601*x^6 +...+ A001006(n)^2*x^n +...
%o A168594 (PARI) {a(n)=if(n==0,1,polcoeff(x/serreverse(x*sum(m=0,n,polcoeff((1/x)*serreverse(x/(1+x+x^2+x^2*O(x^m))), m)^2 *x^m)+x^2*O(x^n)),n))}
%Y A168594 Cf. A001006, A133053, A168344 (variant).
%K A168594 nonn
%O A168594 0,3
%A A168594 _Paul D. Hanna_, Dec 01 2009