A143555 - OEIS

%I #13 Jan 16 2024 17:21:09

%S 1,1,4,8,28,80,308,984,3980,13472,56164,197032,838396,3013872,

%T 13015188,47624568,207971436,771336512,3397886660,12736715592,

%U 56502898140,213618833808,953139545076,3629043226392,16270547827020,62317467147744

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

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

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

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

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

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

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

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

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

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

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

%F (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 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 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 G.f. A(x) = 1 + x + 4*x^2 + 8*x^3 + 28*x^4 + 80*x^5 + 308*x^6 +...

%e A(x)/A(-x) = 1 + 2*x + 2*x^2 + 10*x^3 + 18*x^4 + 98*x^5 + 210*x^6 +...

%e where 1 - (1+x^2)/A(x) = x*A(x)/A(-x).

%e Related expansions:

%e A(x)^2/A(-x)^2 = 1 + 4*x + 8*x^2 + 28*x^3 + 80*x^4 + 308*x^5 +...

%e A(x)^2 = 1 + 2*x + 9*x^2 + 24*x^3 + 88*x^4 + 280*x^5 + 1064*x^6 +...

%e where A(x)^2/A(-x)^2 = A(x)^2 + x + x*A(-x).

%o (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 Cf. A143339, A143554, A143556, A143557, A143558, A143559.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 24 2008