[seqfan] Re: Are A304312, A304313 Same as A006691, A006692?

[Brendan McKay]

It is in my paper files, but I am nearly an earth diameter away and won't
be home for 3 weeks.  Feel free to ask me then.

Brendan.

On 20/5/18 3:14 pm, Neil Sloane wrote:
> I looked hard for the Robinson paper mentioned in A006689-A006692.
> I have many of his papers, but not that one.  I have many conference
> procedings with similar titles, but not that one.
>
> The title of the book in the OEIS entries was rather inaccurate.  Here is a
> corrected reference, giving full details:
>
> Robert W. Robinson, Counting strongly connected finite automata, pages
> 671-685 in "Graph theory with applications to algorithms and computer
> science." Proceedings of the fifth international conference held at Western
> Michigan University, Kalamazoo, Mich., June 4–8, 1984. Edited by Y. Alavi,
> G. Chartrand, L. Lesniak [L. M. Lesniak-Foster], D. R. Lick and C. E. Wall.
> A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1985.
> xv+810 pp. ISBN: 0-471-81635-3; Math Review MR0812651 (86g:05026).
>
> Given this, it should be possible to find the book - maybe Google Books has
> it?
>
>
>
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>
> On Sun, May 20, 2018 at 2:45 AM, Joerg Arndt <arndt at jjj.de> wrote:
>
>> By the argument of "too much of a coincidence" this is very likely:
>> your sequences are built using a functional equation,
>> these tend to count graphy things, and the other
>> sequences do count graphy things in the first place.
>>
>> qed  8-)
>>
>> Best regards,   jj
>>
>> P.S.: I am unable to find the paper online.
>>
>> * Paul Hanna <pauldhanna.math at gmail.com> [May 18. 2018 08:17]:
>>> SeqFans,
>>>         I'd like to share a surprising result, followed by an apparent
>>> connection to finite automata that needs to be shown true/false.
>>>
>>> Consider the power series in x,  A(p,x) ,  such that
>>>      [x^n] exp( n^p * x ) / A(p,x) = 0 for n > 0
>>> where p is a positive integer.
>>>
>>> That is, the coefficient of x^n in  exp( n^p * x ) / A(p,x) equals 0 for
>>> all n > 0.
>>>
>>> At first sight, one would expect A(p,x) to be an e.g.f. with fractional
>>> coefficients.
>>>
>>> The unexpected result is that A(p,x) consists solely of integer
>>> coefficients of x^k, for k>=0 (conjecture 1).
>>> Examples:
>>> https://oeis.org/A304322 (p=2)
>>> https://oeis.org/A304323 (p=3)
>>> https://oeis.org/A304324 (p=4)
>>> https://oeis.org/A304325 (p=5)
>>>
>>> Further, the coefficient of x^n in the logarithmic derivative of A(p,x)
>> wrt
>>> x appears to equal "the number of connected n-state finite automata with
>> p
>>> inputs" (conjecture 2).
>>> If this conjecture holds, then the above conjecture 1 is also true, and
>>> thus A(p,x) consists solely of integer coefficients.
>>> Examples:
>>> https://oeis.org/A304312 (p=2)
>>> https://oeis.org/A304313 (p=3)
>>> https://oeis.org/A304314 (p=4)
>>> https://oeis.org/A304315 (p=5)
>>> These are found in the table https://oeis.org/A304321 of A'(p,x)/A(p,x)
>> for
>>> p >= 1.
>>>
>>> I wonder if someone could show that the following sequences are
>> essentially
>>> the same:
>>> https://oeis.org/A304312  =  https://oeis.org/A006691
>>> https://oeis.org/A304313  =  https://oeis.org/A006692
>>>
>>> The older sequences give a reference on the subject, and perhaps a
>> formula
>>> there could be used to show that these are indeed equivalent.
>>>
>>> Thanks,
>>>        Paul
>>>
>>> --
>>> Seqfan Mailing list - http://list.seqfan.eu/
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
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