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A143555 G.f. satisfies: A(x) = 1 + x*A(x)^2/A(-x)^2.

%I A143555 #13 Jan 16 2024 17:21:09
%S A143555 1,1,4,8,28,80,308,984,3980,13472,56164,197032,838396,3013872,
%T A143555 13015188,47624568,207971436,771336512,3397886660,12736715592,
%U A143555 56502898140,213618833808,953139545076,3629043226392,16270547827020,62317467147744
%N A143555 G.f. satisfies: A(x) = 1 + x*A(x)^2/A(-x)^2.
%C A143555 Specific values:  A(2/9) = 17/9  and  A(-2/9) = 17/18.
%C A143555 Radius of convergence: r = sqrt(2*sqrt(3)-3)/3 = 0.2270833462...
%C A143555 with A(r) = (2 + sqrt(1-3*r))*(1+r^2)/(1+r) = 2.19775350...
%C A143555 and A(-r) = (2 - sqrt(1+3*r))*(1+r^2)/(1-r) = 3*(1+r^2) - A(r) = 0.9569470...
%C A143555 At x=r, the equation (*) (1+x^2)^2 - 2*(1+x^2)*y + (1+x)*y^2 - x*y^3 = 0, which is satisfied by y = A(x), factors out to:  (y - A(r))^2 * (y - A(r)*(1+r^2)/(2*(A(r)-1-r^2))) = 0; this gives the relation: (A(r)-1-r^2)*(3+3*r^2-A(r)) = r*A(r)^2.  At x>r, the equation (*) admits complex solutions for y.
%F A143555 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A143555 (1) (1+x^2)^2 - 2*(1+x^2)*A(x) + (1+x)*A(x)^2 - x*A(x)^3 = 0.
%F A143555 (2) A(x) = 1 + x*A(x)^2 + x^2 + x^2*A(-x).
%F A143555 (3) A(x) = 1 + x^2 + x*A(x)^2/A(-x).
%F A143555 (4) A(x) = 1 + x^2/(1 - A(-x)).
%F A143555 (5) A(x) = 1 + ( 1 - (1+x^2)/A(x) )^2/x.
%F A143555 (6) A(x) = (1+x^2)*G(x) where G(x) = 1 + x*G(x)^2/G(-x) is the g.f. of A143339.
%F A143555 Recurrence: (n-1)*(n+1)*(4*n^3 - 32*n^2 + 71*n - 30)*a(n) = 6*(8*n^3 - 56*n^2 + 101*n - 10)*a(n-1) + 6*(12*n^5 - 132*n^4 + 499*n^3 - 700*n^2 + 102*n + 305)*a(n-2) - 18*(n-4)*(8*n - 25)*a(n-3) + 27*(n-5)*(n-4)*(4*n^3 - 20*n^2 + 19*n + 13)*a(n-4). - _Vaclav Kotesovec_, Dec 29 2013
%F A143555 a(n) ~ c * 3^(n-1) * 2*sqrt(6*sqrt(3)-6 + sqrt(9+6*sqrt(3))) / (2*sqrt(Pi) * (2*sqrt(3)-3)^(n/2+1/4) * n^(3/2)), where c = 4/(2+12^(1/4)) if n is even and c = 12/(6+12^(3/4)) if n is odd. - _Vaclav Kotesovec_, Dec 29 2013
%e A143555 G.f. A(x) = 1 + x + 4*x^2 + 8*x^3 + 28*x^4 + 80*x^5 + 308*x^6 +...
%e A143555 A(x)/A(-x) = 1 + 2*x + 2*x^2 + 10*x^3 + 18*x^4 + 98*x^5 + 210*x^6 +...
%e A143555 where 1 - (1+x^2)/A(x) = x*A(x)/A(-x).
%e A143555 Related expansions:
%e A143555 A(x)^2/A(-x)^2 = 1 + 4*x + 8*x^2 + 28*x^3 + 80*x^4 + 308*x^5 +...
%e A143555 A(x)^2 = 1 + 2*x + 9*x^2 + 24*x^3 + 88*x^4 + 280*x^5 + 1064*x^6 +...
%e A143555 where A(x)^2/A(-x)^2 = A(x)^2 + x + x*A(-x).
%o A143555 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*A^2/subst(A^2,x,-x));polcoeff(A,n)}
%Y A143555 Cf. A143339, A143554, A143556, A143557, A143558, A143559.
%K A143555 nonn
%O A143555 0,3
%A A143555 _Paul D. Hanna_, Aug 24 2008