[seqfan] Re: Conjecture about A127750

[jean-paul allouche]

Dear Allan

You are perfectly right: my remark is totally useless!
As a compensation to my rather stupid remark, and
if you don't have any answer from Paul Barry, I am willing
*WITHOUT being a coauthor* (since I have done
mathematically nothing) to help you converting
some reasonable (and reasonably short) text file
to a LaTeX file -- also agreeing with the suggestion that
you make a pdf for the OEIS, possibly also for ArXiv,
and possibly submit it somewhere afterwards.

best
jp


Le 10/02/2021 à 02:58, Allan Wechsler a écrit :
> Two responses:
>
> First, to Jean-Paul Allouche: I looked up a Hilbert/Cauchy determinant
> explanation, and I think it's unfortunately not applicable in this case,
> though I can absolutely see why you thought of it. In this case, the
> original infinite matrix M is lower-triangular, so the inverse M' is also.
> That means that the "determinant", insofar as the concept applies to
> infinite matrices, is just the product of the main diagonal entries. In the
> case of M, that product is 1 * (1/3) * (1/5) * (1/7) * ..., and the product
> converges rapidly to 0. Similarly, the determinant of M' is the product of
> the main diagonal entries, which is 1 * 3 * 5 * 7 * ... which diverges
> rapidly. So the whole matrix M doesn't really have a well-defined
> determinant, though it is invertible.
>
> Second, to Robert Israel and Neil Sloane: My skills as a LaTician have been
> rusting for more than 30 years. Unless I could get a co-author, I think I'm
> going to have to stick with a text file uploaded to OEIS. (Probably it
> would be most appropriate to co-author with Paul Barry, who first proposed
> the sequence, and I have sent him email with a preliminary query. But I
> don't know if he's still interested in this sequence 13 years later.)
>
> Thanks to everybody; I'm going to crawl off and write now.
>
> On Tue, Feb 9, 2021 at 12:45 PM <israel at math.ubc.ca> wrote:
>
>> You might upload a text file, or even better a nicely formatted .pdf
>> (produced from Latex, for example), in the Links section of the sequence.
>> If you want a more formal publication, you might send it to the Journal of
>> Integer Sequences, and then include a link to that with the sequence
>> (first
>> to the ArXiv preprint, and eventually to the published article).
>>
>> Cheers,
>> Robert
>>
>> On Feb 9 2021, Allan Wechsler wrote:
>>
>>> I'm afraid I'm not familiar with Hilbert determinants and Cauchy
>>> determinants -- but I do have a proof of the stated conjecture, and am
>> just
>>> wondering what to do with it. Even if stated tersely, it wouldn't fit
>>> comfortably in the sequence comments. Do I need to publish a brief paper
>> to
>>> ArXiv and reference it? Or should I upload a text file to OEIS itself? My
>>> main theorem is that A127750(n+1) = 2 * A001151(N) - A209229(N), and the
>>> conjecture follows as an easy corollary.
>>>
>>> On Tue, Feb 9, 2021 at 12:41 AM Jean-Paul Allouche <
>>> jean-paul.allouche at imj-prg.fr> wrote:
>>>
>>>> Hi
>>>>
>>>> Is it conceivable that this determinant has an explicit form (à la
>>>> Hilbert determinant or à la Cauchy determinant)? best jean-paul
>>>>
>>>> Le mar. 9 févr. 2021 à 06:25, Allan Wechsler <acwacw at gmail.com> a
>> écrit :
>>>>> Recently I was investigating a combinatorial system (a simple Turing
>>>>> machine, in fact), and became curious about a sequence it was
>>>> displaying. I
>>>>> looked up the sequence on OEIS and found https://oeis.org/A127750,
>>>>> which matched perfectly.
>>>>>
>>>>> The description in the entry had absolutely nothing to do with my
>>>>> generating system. But there is a conjecture, apparently due to either
>>>> the
>>>>> author, Dr. Paul Barry, or to Neil Sloane -- the entry isn't entirely
>>>>> clear.
>>>>>
>>>>> Obviously I wanted to know if my sequence and Barry's were the same. I
>>>> was
>>>>> able to analyze my own sequence fairly easily, and the conjecture was
>>>> quite
>>>>> trivially true for my sequence. If I could prove the identity of the
>>>>> two sequences, I would have proven the conjecture.
>>>>>
>>>>> Well, now I think I have proven it. What should I do next?
>>>>>
>>>>> --
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>>>>>
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