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

[Joerg Arndt]

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
> 
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