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A191557 G.f. satisfies: A(A(x))^2 = x^2 + 4*A(x)^3.

%I A191557 #10 Mar 30 2012 18:37:26
%S A191557 1,1,1,-1,-5,6,57,-68,-996,1151,23487,-26316,-703858,769268,25912425,
%T A191557 -27791388,-1146924362,1212941187,60112150656,-62911402588,
%U A191557 -3686975047595,3828485422340,262043300715095,-270475215554448,-21394371719691000
%N A191557 G.f. satisfies: A(A(x))^2 = x^2 + 4*A(x)^3.
%C A191557 Compare the g.f. to the following property of G(x) = x*sqrt(1+4*x):
%C A191557 G(G(x))^2 = x^2 + 4*x^3 + 4*G(x)^3.
%F A191557 G.f. satisfies: A(x)^2 = A(-x)^2 + 4*x^3.
%F A191557 G.f. satisfies: A(-A(-x)) = x.
%F A191557 G.f. satisfies: A(x) = B(x) + x^3/B(x) where B(x) = (A(x) - A(-x))/2.
%e A191557 G.f.: A(x) = x + x^2 + x^3 - x^4 - 5*x^5 + 6*x^6 + 57*x^7 - 68*x^8 +...
%e A191557 Note that x^3 is the only odd power of x in A(x)^2:
%e A191557 A(x)^2 = x^2 + 2*x^3 + 3*x^4 - 11*x^6 + 117*x^8 - 2001*x^10 +...
%e A191557 Illustrate A(A(x))^2 = x^2 + 4*A(x)^3 by the expansions:
%e A191557 A(A(x))^2 = x^2 + 4*x^3 + 12*x^4 + 24*x^5 + 16*x^6 - 60*x^7 - 72*x^8 + 640*x^9 + 768*x^10 - 11160*x^11 - 12916*x^12 +...
%e A191557 A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 4*x^6 - 15*x^7 - 18*x^8 + 160*x^9 + 192*x^10 - 2790*x^11 - 3229*x^12 +...
%e A191557 G.f. of odd bisection B(x) = (A(x) - A(-x))/2 begins:
%e A191557 B(x) = x + x^3 - 5*x^5 + 57*x^7 - 996*x^9 + 23487*x^11 +...
%e A191557 where A(x) = B(x) + x^3/B(x).
%o A191557 (PARI) {a(n)=local(A=x+x^2+x*O(x^n));for(i=1,n,A=A-(subst(A,x,A)-x*sqrt(1+4*A^3/x^2))/2);polcoeff(A,n)}
%Y A191557 Cf. A107700, A191565, A138740.
%K A191557 sign
%O A191557 1,5
%A A191557 _Paul D. Hanna_, Jun 06 2011