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

[Paul D Hanna]

SeqFans,  
    Below is my PARI code to generate the coefficients of the series reversion of the functions A(x,m)-1, 
and I put the coefficients in (irregular) triangle form, in case anyone would be interested 
in finding a nice generating method for these polynomials. 
Thanks, 
    Paul 
  
(PARI) {a(n,p)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(p*m)*prod(k=1, m, 1-1/Ser(A)^k)), #A-1)); A[n+1]}
for(n=1,8,print(Vec(serreverse(sum(m=1,n*(n+1)/2,a(m,n)*x^m)+x*O(x^(n*(n+1)/2)))));print("")) 
 
[1]
 
[1, -1, -1]
 
[1, -2, -1, 4, 4, 1]
 
[1, -3, 0, 11, 1, -30, -42, -26, -8, -1]
 
[1, -4, 2, 20, -19, -100, 3, 403, 808, 861, 584, 262, 76, 13, 1]
 
[1, -5, 5, 30, -65, -191, 378, 1557, 103, -8551, -23911, -37958, -41831, -34156, -21179, -10015, -3571, -933, -169, -19, -1]
 
[1, -6, 9, 40, -145, -261, 1384, 2897, -8980, -38710, -14146, 258401, 990407, 2170834, 3426095, 4198850, 4137440, 3336534, 2220430, 1221799, 554027, 205250, 61206, 14351, 2550, 323, 26, 1]
 
[1, -7, 14, 49, -266, -245, 3325, 2596, -36710, -70556, 281645, 1413916, 1184890, -10255248, -54012830, -156371880, -329973512, -552895722, -765517470, -895408431, -896614676, -774834055, -580511469, -377792286, -213512611, -104550572, -44163315, -15985147, -4910774, -1263620, -267378, -45321, -5918, -559, -34, -1]
 
[END] 
---------- Original Message ----------
From: "Paul D Hanna" <pauldhanna at juno.com>
To: seqfan at list.seqfan.eu
Subject: [seqfan] Polynomials in Seres Reversion of a Famiy of Functions
Date: Tue, 26 Jun 2012 04:28:56 GMT

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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