[seqfan] Polynomials in Seres Reversion of a Famiy of Functions

Paul D Hanna pauldhanna at juno.com
Tue Jun 26 06:28:56 CEST 2012


SeqFans, 
   Consider the family of power series A(x,m) that satisfy: 
 
   x = Sum_{n>=1} 1/A(x,m)^(m*n) * Product_{k=1..n} (1 - 1/A(x,m)^k).  
 
We have the following cases in the OEIS: 
m=2: A001002  
m=3: A181997  
m=4: A181998   
m=5: A209441 
m=6: A209442. 
 
 
Now we observe that y = Series_Reversion( A(x,m) - 1 ) is given by the polynomials: 
 
m=1: y = x. 
 
m=2: y = x - x^2 - x^3. 
 
m=3: y = x - 2*x^2 - x^3 + 4*x^4 + 4*x^5 + x^6. 
 
m=4: y = x - 3*x^2 + 11*x^4 + x^5 - 30*x^6 - 42*x^7 - 26*x^8 - 8*x^9 - x^10. 
 
m=5: y = x - 4*x^2 + 2*x^3 + 20*x^4 - 19*x^5 - 100*x^6 + 3*x^7 + 403*x^8 + 808*x^9 + 861*x^10 + 584*x^11 + 262*x^12 + 76*x^13 + 13*x^14 + x^15. 
 
m=6: y = x - 5*x^2 + 5*x^3 + 30*x^4 - 65*x^5 - 191*x^6 + 378*x^7 + 1557*x^8 + 103*x^9 - 8551*x^10 - 23911*x^11 - 37958*x^12 - 41831*x^13 - 34156*x^14 - 21179*x^15 - 10015*x^16 - 3571*x^17 - 933*x^18 - 169*x^19 - 19*x^20 - x^21. 
 
and we suspect the trend to continue. 
 
 
Can anyone find a formula or pattern to these polynomials? 
 
Thanks, 
   Paul 


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