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

Paul Hanna pauldhanna.math at gmail.com
Wed May 16 21:47:31 CEST 2018

```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

```