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

[Neil Sloane]

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/
>
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>