duplicate hunting, pt. 17

[Andrew Plewe]

so at the end of this process (i.e., maybe in July or August) I can
sequence into the search textbox on the OEIS main page to check their OEIS
	-Andrew Plewe-
From: Joshua Zucker [mailto:joshua.zucker at gmail.com]
Sent: Thursday, May 17, 2007 2:54 PM
To: Andrew Plewe
Cc: seqfan at ext.jussieu.fr
Subject: Re: duplicate hunting, pt. 17
sequences (A106742 has offset 0, but that doesn't match the given
Return-Path: <ralf at ark.in-berlin.de>
X-Ids: 168
X-Envelope-From: ralf at ark.in-berlin.de
X-Envelope-To: <seqfan at ext.jussieu.fr>
Date: Fri, 18 May 2007 12:27:34 +0200
From: Ralf Stephan <ralf at ark.in-berlin.de>
To: Sequence Fans <seqfan at ext.jussieu.fr>
Subject: puzzling finding (part.f. and A103446)
Message-ID: <20070518102734.GA16118 at ark.in-berlin.de>
Reply-To: ralf at ark.in-berlin.de
Mail-Followup-To: Sequence Fans <seqfan at ext.jussieu.fr>
MIME-Version: 1.0
Content-Type: text/plain; charset=us-ascii
Content-Disposition: inline
User-Agent: Mutt/1.5.13 (2006-08-11)
X-Greylist: IP, sender and recipient auto-whitelisted, not delayed by milter-greylist-3.0 (shiva.jussieu.fr [134.157.0.168]); Fri, 18 May 2007 12:30:03 +0200 (CEST)
X-Virus-Scanned: ClamAV 0.88.7/3267/Thu May 17 22:40:58 2007 on shiva.jussieu.fr
X-Virus-Status: Clean
X-Miltered: at shiva.jussieu.fr with ID 464D802A.000 by Joe's j-chkmail (http://j-chkmail.ensmp.fr)!

The sequence A103446 has the comment that it is the binomial transform 
of the partition numbers p(1), p(2), ... = 1, 2, 3, 5, 7, 11, 15, 22, 
30, ... (A000041 without the initial 1).

Now, this sequence appears to have a rational o.g.f. of degree 13,
and this puzzles me quite a bit because the partition function is
something without rat. g.f.

Wouldn't it also mean there is a (relatively) simple e.g.f. for
the partition function?

I would be quite interested in a proof of the g.f. or its rationality.
Here is the g.f. starting the sequence with 3,8,...:

(-5*x^12 - 50*x^11 + 277*x^10 - 823*x^9 + 1639*x^8 - 2377*x^7 + 2586*x^6
- 2124*x^5 + 1304*x^4 - 582*x^3 + 179*x^2 - 34*x + 3)/(6*x^13 + 35*x^12
  - 243*x^11 + 790*x^10 - 1691*x^9 + 2628*x^8 - 3078*x^7 + 2751*x^6 -
    1872*x^5 + 955*x^4 - 354*x^3 + 90*x^2 - 14*x + 1)

Regards,
ralf




Ralf, although A103446 refers to my paper with Wieder, 
this sequence is not mentioned in the paper.
It almost certainly does not have a rational o.g.f. of degree 13.
So something is wrong, either with your g.f. or with 
the terms in the sequence.

I would really like to see a description in words of
what A103446 is counting (and also A025168).
I have forgotten how to read the combstruct language!

I will copy this to Thomas Wieder - Thomas, can you
answer these questions?

Neil
>Date: Fri, 18 May 2007 12:27:34 +0200
>From: Ralf Stephan <ralf at ark.in-berlin.de>
>To: Sequence Fans <seqfan at ext.jussieu.fr>
>Subject: puzzling finding (part.f. and A103446)
>
>The sequence A103446 has the comment that it is the binomial transform 
>of the partition numbers p(1), p(2), ... = 1, 2, 3, 5, 7, 11, 15, 22, 
>30, ... (A000041 without the initial 1).
>
>Now, this sequence appears to have a rational o.g.f. of degree 13,
>and this puzzles me quite a bit because the partition function is
>something without rat. g.f.
>
>Wouldn't it also mean there is a (relatively) simple e.g.f. for
>the partition function?
>
>I would be quite interested in a proof of the g.f. or its rationality.
>Here is the g.f. starting the sequence with 3,8,...:
>
>(-5*x^12 - 50*x^11 + 277*x^10 - 823*x^9 + 1639*x^8 - 2377*x^7 + 2586*x^6
>- 2124*x^5 + 1304*x^4 - 582*x^3 + 179*x^2 - 34*x + 3)/(6*x^13 + 35*x^12
>  - 243*x^11 + 790*x^10 - 1691*x^9 + 2628*x^8 - 3078*x^7 + 2751*x^6 -
>    1872*x^5 + 955*x^4 - 354*x^3 + 90*x^2 - 14*x + 1)
>



PS to my initial reply

Ralf, I believe that the terms as shown are correct, but
that your g.f. is wrong.  Here is the defference
between what your gf gives and the true values:

> [seq(s4[i]-t3[i],i=1..40)];
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -31, -498, -5525,


So everything seems to be correct (except your gf!)

Neil