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%I #5 Jul 29 2012 15:14:32
%S 1,2,14,112,910,8008,84588,1059296,13998070,179505848,2193386772,
%T 26007310560,306461781228,3616653947520,42388643986040,
%U 493154764709376,5905712543971814,78382075059128216,1209853310234969668,20945651586098921696,378625571347575985092
%N G.f. satisfies: A(x) = 1/A(-x*A(x)^6).
%C Compare to: W(x) = 1/W(-x*W(x)^6) when W(x) = Sum_{n>=0} (3*n+1)^(n-1)*x^n/n!.
%C An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^6); for example, (*) is satisfied by G(x) = W(m*x), where W(x) = Sum_{n>=0} (3*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)^6))/2 starting at G_0(x) = 1+2*x.
%e G.f.: A(x) = 1 + 2*x + 14*x^2 + 112*x^3 + 910*x^4 + 8008*x^5 + 84588*x^6 +...
%e A(x)^6 = 1 + 12*x + 144*x^2 + 1672*x^3 + 18720*x^4 + 207000*x^5 + 2339072*x^6 +...
%o (PARI) {a(n)=local(A=1+2*x);for(i=0,n,A=(A+1/subst(A,x,-x*A^6+x*O(x^n)))/2);polcoeff(A,n)}
%o for(n=0,31,print1(a(n),", "))
%Y Cf. A214761, A214762, A214763, A214764, A214765, A214767, A214768, A214769.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 29 2012
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