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A143339
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G.f. satisfies: A(x) = 1 + x*A(x)^2/A(-x).
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9
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1, 1, 3, 7, 25, 73, 283, 911, 3697, 12561, 52467, 184471, 785929, 2829401, 12229259, 44795167, 195742177, 726541345, 3202144483, 12010174247, 53300753657, 201608659561, 899838791419, 3427434566831, 15370709035601, 58890032580913
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OFFSET
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0,3
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COMMENTS
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Specific values: A(2/9) = 9/5 and A(-2/9) = 9/10.
Radius of convergence: r = sqrt(2*sqrt(3)-3)/3 = 0.2270833462...
with A(r) = (2 + sqrt(1-3*r))/(1+r) = 2.0899798397...
and A(-r) = (2 - sqrt(1+3*r))/(1-r) = 3 - A(r) = r*A(r)^2/(A(r)-1) = 0.91002016...
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.
The limit a(n+1)/a(n) does not exist but oscillates between 2 attractors:
Limit a(2*n)/a(2*n-1) = sqrt(3)+3, Limit a(2*n+1)/a(2*n) = 3*(sqrt(3)+1)/2.
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 1 - 2*A(x) + (1+x)*A(x)^2 - (x+x^3)*A(x)^3 = 0.
(2) A(x) = exp( Sum_{n>=1} A(x)^n / A(-x)^n * x^n/n ).
(3) A(x) = Sum_{n>=0} x^n * A(x)^n / A(-x)^n.
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
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EXAMPLE
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A bisection of g.f. A(x) equals a bisection of A(x)^2:
A(x) = 1 + x + 3*x^2 + 7*x^3 + 25*x^4 + 73*x^5 + 283*x^6 + 911*x^7 +...
A(x)^2 = 1 + 2*x + 7*x^2 + 20*x^3 + 73*x^4 + 238*x^5 + 911*x^6 +...
that is, A(x) - x*A(x)^2 = 1 + x^2*A(x)*A(-x), where
A(x)*A(-x) = 1 + 5*x^2 + 45*x^4 + 521*x^6 + 6873*x^8 + 98061*x^10 +...
Related expressions:
A(x) = 1 + x*A(x)/A(-x) + x^2*A(x)^2/A(-x)^2 + x^3*A(x)^3/A(-x)^3 +...
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 +...
Illustrate the behavior of a(n+1)/a(n) as n grows:
a(301)/a(300) = 4.07522764...
a(302)/a(301) = 4.71149410...
a(303)/a(302) = 4.07537802...
a(304)/a(303) = 4.71162882...
the limits of which approach the attractors:
3*(sqrt(3)+1)/2 = 4.09807621... and sqrt(3)+3 = 4.73205080...
note that the product of the attractors equals 1/r^2, where
r = sqrt(2*sqrt(3)-3)/3 = sqrt(2/sqrt(3))/(sqrt(3)+3)
is the radius of convergence of the g.f. A(x).
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MATHEMATICA
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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 *)
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PROG
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(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)}
(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)}
(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)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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