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

[Olivier Gerard]

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]
>
> [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]
>
>