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%I #5 Jul 29 2012 15:14:52
%S 1,2,18,180,1734,18300,270420,5151720,96203910,1565102844,22108977596,
%T 287976684088,3835267955036,55283720348664,804522994149032,
%U 10849701753955856,150403977728200774,3086256025416536700,91156710989444409004,2687925748932854737432
%N G.f. satisfies: A(x) = 1/A(-x*A(x)^8).
%C Compare to: W(x) = 1/W(-x*W(x)^8) when W(x) = Sum_{n>=0} (4*n+1)^(n-1)*x^n/n!.
%C An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^8); for example, (*) is satisfied by G(x) = W(m*x), where W(x) = Sum_{n>=0} (4*n+1)^(n-1)*x^n/n!.
%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)^8))/2 starting at G_0(x) = 1+2*x.
%e G.f.: A(x) = 1 + 2*x + 18*x^2 + 180*x^3 + 1734*x^4 + 18300*x^5 + 270420*x^6 +...
%e A(x)^8 = 1 + 16*x + 256*x^2 + 3904*x^3 + 56320*x^4 + 793984*x^5 + 11567104*x^6 +...
%o (PARI) {a(n)=local(A=1+2*x);for(i=0,n,A=(A+1/subst(A,x,-x*A^8+x*O(x^n)))/2);polcoeff(A,n)}
%o for(n=0,31,print1(a(n),", "))
%Y Cf. A214761, A214762, A214763, A214764, A214765, A214766, A214767, A214769.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 29 2012
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