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%I #3 Jul 29 2012 14:37:08
%S 1,2,16,144,1280,12416,156288,2445952,39005696,569584128,7551139840,
%T 94905663488,1200235880448,15657039026176,204235121909760,
%U 2589347043356672,34080849916796928,554466780012625920,11679936697324273664,269604415927633805312,6025264829519275556864
%N G.f. satisfies: A(x) = 1/A(-x*A(x)^7).
%C Compare g.f. to: G(x) = 1/G(-x*G(x)^7) when G(x) = 1 + x*G(x)^4 (A002293).
%C An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^7); for example, (*) is satisfied by G(x) = F(m*x) = 1 + m*x*F(m*x)^4 for all m, where F(x) is the g.f. of A002293.
%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)^7))/2 starting at G_0(x) = 1+2*x.
%e G.f.: A(x) = 1 + 2*x + 16*x^2 + 144*x^3 + 1280*x^4 + 12416*x^5 + 156288*x^6 +...
%e A(x)^4 = 1 + 8*x + 88*x^2 + 992*x^3 + 10896*x^4 + 121600*x^5 + 1492480*x^6 +...
%e A(x)^7 = 1 + 14*x + 196*x^2 + 2632*x^3 + 33712*x^4 + 424032*x^5 + 5484864*x^6 +...
%o (PARI) {a(n)=local(A=1+2*x);for(i=0,n,A=(A+1/subst(A,x,-x*A^7+x*O(x^n)))/2);polcoeff(A,n)}
%o for(n=0,31,print1(a(n),", "))
%Y Cf. A214761, A214762, A214763, A214764, A214765, A214766, A214768, A214769.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 29 2012
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