[seqfan] Re: Constant Arising from a Certain Functional Eq

[Paul D Hanna]

Seqfans, 
      Let me demonstrate how I arrived at the value for: 
the least t such that F(x,t) consists entirely of non-negative coefficients where 
    F(x,t) = (1+x)^2 + t*x^2*F'(x,t)/F(x,t). 
 
Perhaps this constant (label it r) would lend itself more easily to analysis. 
 
The constant equals the limit of the maximum real root (when it exists) to the coefficient polynomials in F(x,t), where  
    F(x,t) = Sum_{n>=0} C(n,t)*x^n 
and so r = limit maximum real root to C(n,t) = 0. 
 
The coefficient polynomials begin: 
1;
2;
2*t + 1;
4*t^2 - 2*t;
12*t^3 - 18*t^2 + 2*t;
48*t^4 - 112*t^3 + 48*t^2 - 2*t;
240*t^5 - 720*t^4 + 540*t^3 - 100*t^2 + 2*t;
1440*t^6 - 5088*t^5 + 5272*t^4 - 1848*t^3 + 180*t^2 - 2*t;
10080*t^7 - 39984*t^6 + 51240*t^5 - 26124*t^4 + 5096*t^3 - 294*t^2 + 2*t;
80640*t^8 - 348864*t^7 + 519936*t^6 - 340048*t^5 + 100320*t^4 - 12096*t^3 + 448*t^2 - 2*t; 
725760*t^9 - 3360960*t^8 + 5610144*t^7 - 4384800*t^6 + 1707768*t^5 - 321156*t^4 + 25704*t^3 - 648*t^2 + 2*t; ...
  
If one could find a formula for these coefficients, then perhaps a CF or other expression for the constant r could be found. 
 
My PARI code: 
F=(1+x)^2;for(i=1,121,F=(1+x)^2 + t*x^2*F'/(F+O(x^121))) 
for(n=10,120,print(solve(x=1.5,2.1,subst(polcoeff(F,n),t,x))))
  
The maximum real root to C(n,t) = 0 then converges to 
r = 2.044417702215325822734054118424444123239275900404337673654769113888960412061213453842830399116759613119256808091804173623881399520091186311515725391346422605725921559292857003581776...
  
Thank you, 
    Paul