# [seqfan] Re: Polynomials in Seres Reversion of a Famiy of Function s

Paul D Hanna pauldhanna at juno.com
Wed Jul 25 07:26:54 CEST 2012

```Hello Gerard,
Your hints were sufficient to arrive at the g.f. of the new triangle A214670:

A(x,y) = Sum_{n>=1} -x^n * Product_{k=1..n} (1 - (1+y)^k) / (1 - x*(1+y)^k)

in which row n is the finite polynomial P(y,n) defined by:
A(x,y) = Sum_{n>=1} x^n * P(y,n)
such that:
P(y,n) = Series_Reversion( G(y,n) )
where G(y,n) satisfies:
y = Sum_{m>=1} 1/G(y,n)^(n*m) * Product_{k=1..n} (1 - 1/G(y,n)^k),
for n>=1.

Paul

---------- Original Message ----------
From: Olivier Gerard <olivier.gerard at gmail.com>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: Polynomials in Seres Reversion of a Famiy of Functions
Date: Tue, 26 Jun 2012 21:09:20 +0200

Dear Paul,

I find your investigations very interesting.

You have certainly remarked that the sum of coefficients

1, *1*, *7*, *97*, *2911*, 180481, 22740607, 5776114177,

appears to be  A005014 which is an inverse binomial transform.

This would suggest to look at decompositions of

Sum[t^n*Product[(2^i - 1)/(-1 + 2^i*t), {i, 1, n}], {n, 0, Infinity}]

(whose series expansion match this sum, including the signs)

as the basis for producing directly the polynomials by introducing at least
another variable in it.
The link to nxn binary matrices and graphs is another valuable track to
follow.

Olivier

On Tue, Jun 26, 2012 at 6:27 PM, Paul D Hanna <pauldhanna at juno.com> wrote:

> 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, -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]
>
>

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