A143339 G.f. satisfies: A(x) = 1 + x*A(x)^2/A(-x).
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
Offset: 0
Keywords
Examples
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).
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..300
Programs
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Mathematica
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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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)}
Formula
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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