%I A088717
%S A088717 1,1,3,14,84,596,4785,42349,406287,4176971,45640572,526788153,
%T A088717 6392402793,81247489335,1078331283648,14907041720241,214187010762831,
%U A088717 3192620516380376,49287883925072010,786925082232918304
%N A088717 G.f. satisfies: A(x) = 1 + x*A(x)^2*A(x*A(x)^2).
%C A088717 Coefficient of x^n of A(x)^2 equals coefficient of x^n in (1+x*A(x))^(n+1)/
               (n+1).
%F A088717 Suppose functions A=A(x), B=B(x), C=C(x), etc., satisfy: A = 1 + xA^2*B, 
               B = 1 + x(AB)^2*C, C = 1 + x(ABC)^2*D, D = 1 + x(ABCD)^2*E, etc., 
               then B(x)=A(x*A(x)^2), C(x)=B(x*A(x)^2), D(x)=C(x*A(x)^2), etc., 
               where A(x) = 1 + x*A(x)^2*A(x*A(x)^2) is the g.f. of this sequence 
               (see table A128330).
%F A088717 G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,
               n)*F(x,n+1)) for n>0 with F(x,0)=1. - Paul D. Hanna (pauldhanna(AT)juno.com), 
               Apr 16 2007
%e A088717 Comment from Paul D. Hanna (pauldhanna(AT)juno.com), Apr 16 2007: G.f. 
               A(x) is the unique solution to variable A in the infinite system 
               of simultaneous equations:
%e A088717 A = 1 + xAB;
%e A088717 B = A*(1 + xBC);
%e A088717 C = B*(1 + xCD);
%e A088717 D = C*(1 + xDE);
%e A088717 E = D*(1 + xEF) ; ...
%o A088717 (PARI) {a(n)=local(A=1+x);for(i=0,n,A=1+x*A^2*subst(A,x,x*A^2+x*O(x^n)));
               polcoeff(A,n)}
%Y A088717 Cf. A128330, A030266.
%Y A088717 Cf. A002449, A030266, A087949, A088714, A091713, A120971.
%Y A088717 Sequence in context: A154757 A074535 A005700 this_sequence A111538 A088716 
               A005189
%Y A088717 Adjacent sequences: A088714 A088715 A088716 this_sequence A088718 A088719 
               A088720
%K A088717 nonn,eigen
%O A088717 0,3
%A A088717 Paul D. Hanna (pauldhanna(AT)juno.com), Oct 12 2003 and Mar 10 2007