[seqfan] Exponents in a Functional Equation Yielding Nonnegative Series Solutio n

Paul D Hanna pauldhanna at juno.com
Sun Nov 27 20:52:53 CET 2011

     Could someone provide some initial terms to the following sequence. 
Largest integer exponent a(n) in the functional equation: 
     F(x,n) = 1 + x*F(x,n)^n + x^2 / F(x,n)^a(n) 
yielding a nonnegative integer power series solution F(x,n), n>=1. 
Here a nonnegative integer power series is a power series in x, F(x) = Sum_{k>=0} c(k)*x^k, where coefficients c(k) >= 0 for all k >= 0. 
I get the following initial terms (offset 1): 
these terms could be wrong; it seems difficult to establish that a power series solution F(x,n) consists entirely of nonnegative coefficients. 
Once verified, the coefficients in the respective optimal power series solutions to F(x,n) for n=2,3,4,5,6, for the above functional equation could be worthy of submitting to OEIS:  
n=2: [1,1,3,4,11,53,146,366,1563,5837,16720,59917,246090,...]; 
n=3: [1,1,4,6,52,393,1008,6148,61860,218676,974226,11400698,53140164,...]; 
n=4: [1,1,5,11,153,1310,4393,53450,572721,2381127,24825691,304983749,...]; 
n=5: [1,1,6,18,341,3226,13783,264089,2965655,14883544,273437516,3387026011,...]; 
n=6: [1,1,7,27,647,6602,34923,945048,10812285,66308487,1856333009,22217696338,...] 
which are not in the OEIS. 
Finally, it seems that there exists maximal real-valued constants r(n) >=0 such that 
     F(x,n) = 1 + x*F(x,n)^n + x^2/F(x,n)^R 
has a nonnegative power series solution F(x,n) for R <= r(n); 
the above sequence would equal the floor function of these constants r(n). 
I would like very much to know the approximate value of these constants r(n) for n>=1. 

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