duplicate hunting, pt. 17
Andrew Plewe
aplewe at sbcglobal.net
Fri May 18 00:13:08 CEST 2007
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
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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)
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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
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