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%I #12 Sep 08 2013 16:04:38
%S 1,1,6,41,334,2901,26651,253709,2483395,24829132,252506507,2603798287,
%T 27161758393,286118173600,3039211373800,32517513415886,
%U 350122302629869,3790909121211262,41249405668333107,450832515809731316,4947009705400704588,54479711308604703264,601933495810972446631
%N G.f. A(x) satisfies: 1/A(x)^2 + 4*x*A(x)^2 = 1/A(x^2) + 2*x*A(x^2).
%F G.f. A(x) satisfies:
%F (1) 1/A(x)^2 + 4*x*A(x)^2 = F(x)^2,
%F (2) 1/A(x^2) + 2*x*A(x^2) = F(x)^2,
%F (3) A(x) = ( (F(x)^2 - sqrt(F(x)^4 - 16*x)) / (8*x) )^(1/2),
%F (4) A(x^2) = (F(x)^2 - sqrt(F(x)^4 - 8*x)) / (4*x),
%F where F(x) = (F(x^2)^2 + 4*x)^(1/4) is the g.f. of A107086.
%e G.f.: A(x) = 1 + x + 6*x^2 + 41*x^3 + 334*x^4 + 2901*x^5 + 26651*x^6 +...
%e such that A(x) satisfies the identity illustrated by:
%e 1/A(x)^2 + 4*x*A(x)^2 = 1 + 2*x - x^2 + 2*x^3 - 5*x^4 + 12*x^5 - 30*x^6 +...
%e 1/A(x^2) + 2*x*A(x^2) = 1 + 2*x - x^2 + 2*x^3 - 5*x^4 + 12*x^5 - 30*x^6 +...
%e Related expansions.
%e A(x)^2 = 1 + 2*x + 13*x^2 + 94*x^3 + 786*x^4 + 6962*x^5 + 64793*x^6 +...
%e A(x)^4 = 1 + 4*x + 30*x^2 + 240*x^3 + 2117*x^4 + 19512*x^5 + 186706*x^6 +...
%e 1/A(x) = 1 - x - 5*x^2 - 30*x^3 - 233*x^4 - 1949*x^5 - 17503*x^6 +...
%e 1/A(x)^2 = 1 - 2*x - 9*x^2 - 50*x^3 - 381*x^4 - 3132*x^5 - 27878*x^6 +...
%e The g.f. of A107086 begins:
%e F(x) = 1 + x - x^2 + 2*x^3 - 5*x^4 + 13*x^5 - 35*x^6 + 99*x^7 - 289*x^8 +...
%e where F(x)^4 = F(x^2)^2 + 4*x:
%e F(x)^2 = 1 + 2*x - x^2 + 2*x^3 - 5*x^4 + 12*x^5 - 30*x^6 + 82*x^7 - 233*x^8 +...
%e F(x)^4 = 1 + 4*x + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 + 82*x^14 +...
%o (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1/subst(A, x, x^2) + 2*x*subst(A, x, x^2) - 4*x*A^2 +x*O(x^n))^(1/2)); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A107086.
%Y Cf. variants: A228712, A228928.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Aug 30 2013
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