%I #5 Jul 29 2012 15:20:04
%S 1,2,12,84,616,4832,42112,410368,4316800,46899648,512004480,
%T 5554843904,59657443584,633013100288,6639969848320,69332566233088,
%U 733169635126272,8068863012833280,95049764691595264,1213724245095528448,16619899465108049920,238054738089559379968
%N G.f. satisfies: A(x) = 1/A(-x*A(x)^5).
%C Compare g.f. to: G(x) = 1/G(-x*G(x)^5) when G(x) = 1 + x*G(x)^3 (A001764).
%C An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^5); for example, (*) is satisfied by G(x) = F(m*x) = 1 + m*x*F(m*x)^3 for all m, where F(x) is the g.f. of A001764.
%F The g.f. of this sequence is the limit of the recurrence:
%F (*) G_{n+1}(x) = (G_n(x) + 1/G_n(-x*G_n(x)^5))/2 starting at G_0(x) = 1+2*x.
%e G.f.: A(x) = 1 + 2*x + 12*x^2 + 84*x^3 + 616*x^4 + 4832*x^5 + 42112*x^6 +...
%e A(x)^3 = 1 + 6*x + 48*x^2 + 404*x^3 + 3432*x^4 + 29808*x^5 + 271056*x^6 +...
%e A(x)^5 = 1 + 10*x + 100*x^2 + 980*x^3 + 9400*x^4 + 89632*x^5 + 866080*x^6 +...
%o (PARI) {a(n)=local(A=1+2*x);for(i=0,n,A=(A+1/subst(A,x,-x*A^5+x*O(x^n)))/2);polcoeff(A,n)}
%o for(n=0,31,print1(a(n),", "))
%Y Cf. A214761, A214762, A214763, A214764, A214766, A214767, A214768, A214769.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 29 2012