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

1, 1, 4, 8, 28, 80, 308, 984, 3980, 13472, 56164, 197032, 838396, 3013872, 13015188, 47624568, 207971436, 771336512, 3397886660, 12736715592, 56502898140, 213618833808, 953139545076, 3629043226392, 16270547827020, 62317467147744

Offset: 0

Author

Paul D. Hanna, Aug 24 2008

Keywords

• nonn

Comments

Specific values: A(2/9) = 17/9 and A(-2/9) = 17/18.

Radius of convergence: r = sqrt(2*sqrt(3)-3)/3 = 0.2270833462...

with A(r) = (2 + sqrt(1-3*r))*(1+r^2)/(1+r) = 2.19775350...

and A(-r) = (2 - sqrt(1+3*r))*(1+r^2)/(1-r) = 3*(1+r^2) - A(r) = 0.9569470...

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.

Examples

			G.f. A(x) = 1 + x + 4*x^2 + 8*x^3 + 28*x^4 + 80*x^5 + 308*x^6 +...
A(x)/A(-x) = 1 + 2*x + 2*x^2 + 10*x^3 + 18*x^4 + 98*x^5 + 210*x^6 +...
where 1 - (1+x^2)/A(x) = x*A(x)/A(-x).
Related expansions:
A(x)^2/A(-x)^2 = 1 + 4*x + 8*x^2 + 28*x^3 + 80*x^4 + 308*x^5 +...
A(x)^2 = 1 + 2*x + 9*x^2 + 24*x^3 + 88*x^4 + 280*x^5 + 1064*x^6 +...
where A(x)^2/A(-x)^2 = A(x)^2 + x + x*A(-x).
		

Crossrefs

Cf. A143339, A143554, A143556, A143557, A143558, A143559.

Programs

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)}

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) (1+x^2)^2 - 2*(1+x^2)*A(x) + (1+x)*A(x)^2 - x*A(x)^3 = 0.

(2) A(x) = 1 + x*A(x)^2 + x^2 + x^2*A(-x).

(3) A(x) = 1 + x^2 + x*A(x)^2/A(-x).

(4) A(x) = 1 + x^2/(1 - A(-x)).

(5) A(x) = 1 + ( 1 - (1+x^2)/A(x) )^2/x.

(6) A(x) = (1+x^2)*G(x) where G(x) = 1 + x*G(x)^2/G(-x) is the g.f. of A143339.

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

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