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%I #48 Jan 22 2024 19:38:25
%S 1,1,3,7,25,73,283,911,3697,12561,52467,184471,785929,2829401,
%T 12229259,44795167,195742177,726541345,3202144483,12010174247,
%U 53300753657,201608659561,899838791419,3427434566831,15370709035601,58890032580913
%N G.f. satisfies: A(x) = 1 + x*A(x)^2/A(-x).
%C Specific values: A(2/9) = 9/5 and A(-2/9) = 9/10.
%C Radius of convergence: r = sqrt(2*sqrt(3)-3)/3 = 0.2270833462...
%C with A(r) = (2 + sqrt(1-3*r))/(1+r) = 2.0899798397...
%C and A(-r) = (2 - sqrt(1+3*r))/(1-r) = 3 - A(r) = r*A(r)^2/(A(r)-1) = 0.91002016...
%C At x=r, the equation (*) 1 - 2*y + (1+x)*y^2 - (x+x^3)*y^3 = 0, which is satisfied by y = A(x), factors out to: (y - A(r))^2 * (y - A(r)/(2*A(r)-2)) = 0; this gives the relation: (A(r)-1)*(3-A(r))/A(r)^2 = r. At x>r, the equation (*) admits complex solutions for y.
%C The limit a(n+1)/a(n) does not exist but oscillates between 2 attractors:
%C Limit a(2*n)/a(2*n-1) = sqrt(3)+3, Limit a(2*n+1)/a(2*n) = 3*(sqrt(3)+1)/2.
%H Paul D. Hanna, <a href="/A143339/b143339.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F (1) 1 - 2*A(x) + (1+x)*A(x)^2 - (x+x^3)*A(x)^3 = 0.
%F (2) A(x) = exp( Sum_{n>=1} A(x)^n / A(-x)^n * x^n/n ).
%F (3) A(x) = Sum_{n>=0} x^n * A(x)^n / A(-x)^n.
%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) + (68*n^5 - 760*n^4 + 2927*n^3 - 4202*n^2 + 683*n + 1800)*a(n-2) + 12*(4*n^3 - 40*n^2 + 136*n - 155)*a(n-3) + 3*(60*n^5 - 768*n^4 + 3509*n^3 - 6422*n^2 + 2571*n + 2950)*a(n-4) - 18*(n-4)*(8*n - 25)*a(n-5) + 27*(n-5)*(n-4)*(4*n^3 - 20*n^2 + 19*n + 13)*a(n-6). - _Vaclav Kotesovec_, Feb 17 2014
%e A bisection of g.f. A(x) equals a bisection of A(x)^2:
%e A(x) = 1 + x + 3*x^2 + 7*x^3 + 25*x^4 + 73*x^5 + 283*x^6 + 911*x^7 +...
%e A(x)^2 = 1 + 2*x + 7*x^2 + 20*x^3 + 73*x^4 + 238*x^5 + 911*x^6 +...
%e that is, A(x) - x*A(x)^2 = 1 + x^2*A(x)*A(-x), where
%e A(x)*A(-x) = 1 + 5*x^2 + 45*x^4 + 521*x^6 + 6873*x^8 + 98061*x^10 +...
%e Related expressions:
%e A(x) = 1 + x*A(x)/A(-x) + x^2*A(x)^2/A(-x)^2 + x^3*A(x)^3/A(-x)^3 +...
%e log(A(x)) = A(x)/A(-x)*x + A(x)^2/A(-x)^2*x^2/2 + A(x)^3/A(-x)^3*x^3/3 +...
%e Illustrate the behavior of a(n+1)/a(n) as n grows:
%e a(301)/a(300) = 4.07522764...
%e a(302)/a(301) = 4.71149410...
%e a(303)/a(302) = 4.07537802...
%e a(304)/a(303) = 4.71162882...
%e the limits of which approach the attractors:
%e 3*(sqrt(3)+1)/2 = 4.09807621... and sqrt(3)+3 = 4.73205080...
%e note that the product of the attractors equals 1/r^2, where
%e r = sqrt(2*sqrt(3)-3)/3 = sqrt(2/sqrt(3))/(sqrt(3)+3)
%e is the radius of convergence of the g.f. A(x).
%t terms = 26; A[_] = 1; Do[A[x_] = 1 + x*A[x]^2/A[-x] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* _Jean-François Alcover_, Jan 15 2018 *)
%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*A^2/subst(A,x,-x));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(sum(m=1,n,A^m/subst(A^m,x,-x+x*O(x^n))*x^m/m)));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=sum(m=0,n,x^m*A^m/subst(A^m,x,-x+x*O(x^n))));polcoeff(A,n)}
%Y Cf. A143562, A216681, A212527, A216713.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Aug 09 2008
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