The g.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion( x*(1-x)*(3 - 2*x + sqrt(1+4*x-4*x^2))/4 );
(2) (A(x) - A(-x))/2 = x*(A(x)^2 + A(-x)^2)/2;
(3) ((A(x) + A(-x))/2)^3 = F(x^2), where F(x) = (1/x)*Series_Reversion( x*(1+x)^3/(1+2*x)^6 );
(4) 1 - x*(A(x) - A(-x))/2 = x/Series_Reversion( x - x*(C(x) + C(-x))/2 ), where C(x) = (1 - sqrt(1-4*x))/2 is the Catalan function (A000108);
(5a) (1/A(x) + 1/A(-x))/2 = ( 1 - x*(A(x) - A(-x))/2 )^2;
(5b) (1/A(x) - 1/A(-x))/2 = (-x)/(1 - 2*x*(A(x) - A(-x))/2);
(6a) (1/A(x)^2 + 1/A(-x)^2)/2 = ( 1 - x*(A(x) - A(-x))/2 )^3.
(6b) (1/A(x)^2 - 1/A(-x)^2)/2 = -2*x*(1 - x*(A(x) - A(-x))/2)^2/( 1 - x*(A(x) - A(-x)) ).
Let B(x) = Series_Reversion( x*(1-x^2)/(1+x^2)^3 ), then
(7) A(x) = (1 - sqrt(1 - 4*x - 4*x*B(x)^2))/(2*x);
(8) A(x) - x*A(x)^2 = A(-x) + x*A(-x)^2 = 1 + B(x)^2;
(9) 1 - x*(A(x) - A(-x))/2 = 1/(1 + B(x)^2);
(10) 1/A(x) = 1/(1 + B(x)^2)^2 - x*(1 + B(x)^2)/(1 - B(x)^2);
(10a) (1/A(x) + 1/A(-x))/2 = 1/(1 + B(x)^2)^2;
(10b) (1/A(x) - 1/A(-x))/2 = (-x)*(1 + B(x)^2)/(1 - B(x)^2);
(11) 1/A(x)^2 = 1/(1 + B(x)^2)^3 - 2*x/(1 - B(x)^4);
(11a) (1/A(x)^2 + 1/A(-x)^2)/2 = 1/(1 + B(x)^2)^3;
(11b) (1/A(x)^2 - 1/A(-x)^2)/2 = -2*x/(1 - B(x)^4).
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