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A214763 G.f. satisfies: A(x) = 1/A(-x*A(x)^3).

%I A214763 #11 Nov 12 2016 04:36:13
%S A214763 1,2,8,40,224,1280,7168,40000,231296,1436928,9773056,72242176,
%T A214763 563679232,4491707904,35735001088,280941652992,2178641254400,
%U A214763 16710771339264,127402021142528,970887186407424,7436390169329664,57531833133899776,451525691751628800,3608174274928951296
%N A214763 G.f. satisfies: A(x) = 1/A(-x*A(x)^3).
%C A214763 Compare to: G(x) = 1/G(-x*G(x)^3) when G(x) = 1 + x*G(x)^2 (A000108).
%C A214763 Compare to: B(x) = 1/B(-x*B(x)^3) when B(x) = 1/(1-9*x)^(1/3) = g.f. of A004987.
%C A214763 An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^3); for example, (*) is satisfied by G(x) = C(m*x) = (1-sqrt(1-4*m*x))/(2*m*x) for all m, where C(x) is the Catalan function.
%H A214763 Paul D. Hanna, <a href="/A214763/b214763.txt">Table of n, a(n) for n = 0..200</a>
%F A214763 The g.f. of this sequence is the limit of the recurrence:
%F A214763 (*) G_{n+1}(x) = (G_n(x) + 1/G_n(-x*G_n(x)^3))/2 starting at G_0(x) = 1+2*x.
%e A214763 G.f.: A(x) = 1 + 2*x + 8*x^2 + 40*x^3 + 224*x^4 + 1280*x^5 + 7168*x^6 +...
%e A214763 A(x)^2 = 1 + 4*x + 20*x^2 + 112*x^3 + 672*x^4 + 4096*x^5 + 24640*x^6 +...
%e A214763 A(x)^3 = 1 + 6*x + 36*x^2 + 224*x^3 + 1440*x^4 + 9312*x^5 + 59456*x^6 +...
%e A214763 1/A(x) = A(-x*A(x)^3) = 1 - 2*x - 4*x^2 - 16*x^3 - 80*x^4 - 384*x^5 - 1664*x^6 - 7360*x^7 - 40832*x^8 - 304128*x^9 - 2667008*x^10 -...
%o A214763 (PARI) {a(n)=local(A=1+2*x);for(i=0,n,A=(A+1/subst(A,x,-x*A^3+x*O(x^n)))/2);polcoeff(A,n)}
%o A214763 for(n=0,30,print1(a(n),", "))
%Y A214763 Cf. A214761, A214762, A214764, A214765, A214766, A214767, A214768, A214769.
%K A214763 nonn
%O A214763 0,2
%A A214763 _Paul D. Hanna_, Jul 27 2012