%I #6 Mar 30 2012 18:37:26
%S 1,2,-14,184,-3194,65472,-1503924,37593664,-1004163802,28314667072,
%T -835650200380,25652840146624,-815280469973380,26728163562423360,
%U -901336722528156712,31194183364269262848,-1105930698812430437626
%N G.f. satisfies: A(A(x))^2 = A(x)^2 + 4*x^3.
%e G.f.: A(x) = x + 2*x^2 - 14*x^3 + 184*x^4 - 3194*x^5 + 65472*x^6 +...
%e Illustrate A(A(x))^2 - A(x)^2 = 4*x^3 with the expansions:
%e A(x)^2 = x^2 + 4*x^3 - 24*x^4 + 312*x^5 - 5456*x^6 + 113016*x^7 +...
%e A(A(x))^2 = x^2 + 8*x^3 - 24*x^4 + 312*x^5 - 5456*x^6 + 113016*x^7 +...
%e A(A(x)) = x + 4*x^2 - 20*x^3 + 236*x^4 - 3872*x^5 + 76716*x^6 - 1723488*x^7 +...
%e Let R(x) be the series reversion of A(x):
%e R(x) = x - 2*x^2 + 22*x^3 - 364*x^4 + 7390*x^5 - 170556*x^6 +...
%e R(x)^3 = x^3 - 6*x^4 + 78*x^5 - 1364*x^6 + 28254*x^7 - 655668*x^8 +...
%e where A(x)^2 = x^2 + 4*R(x)^3.
%o (PARI) {a(n)=local(A=x+x^2+x*O(x^n));for(i=1,n,A=A-(subst(A,x,A)-x*sqrt(4*x+A^2/x^2)));polcoeff(A,n)}
%Y Cf. A191557, 107700, A138740.
%K sign
%O 1,2
%A _Paul D. Hanna_, Jun 06 2011