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

[Max Alekseyev]

Hi Paul,

First, when you say "one would expect A(p,x) to be an e.g.f. with
fractional coefficients" - it seems that you mean o.g.f. not e.g.f. here.
One can easily show that the the e.g.f. indeed have integer coefficients
(see below), but it's quite surprising that they are divisible by the
corresponding factorials and thus form an o.g.f. with integer coefficients.

Second, one can derive an explicit recurrence for log(A(p,x)) as follows.

Let us fix a positive integer p, and denote f(x) = log(A(p,x)). Then the
defining identity for A(p,x) can be stated as
(*)    [x^n] exp( n^p * x - f(x) ) = 0 for all integers n > 0.

Let f_i/i! be the coefficient of x^i in f(x). Then (*) translates into
a recurrence formula:
f_0 = 0,
f_1 = 1,
and for n>1,
f_n = B_n(n^p-f_1,-f_2,...,-f_{n-1},0).
It further implies that all f_n are integer. And so are the e.g.f.
coefficients of A(p,x) = exp(f(x)).

E.g., for https://oeis.org/A304312 (p=2) as the derivative of log(A(2,x)),
we have A304312(n) = f_{n+1}/n! and thus the formula:

A304312(0) = 1,
and for n>=1,
A304312(n) = B_{n+1}((n+1)^2-0!*A304312(0),-1!*A304312(1),...,-(n-1)!*
A304312(n-1),0) / n!.

This still however does not explain why the resulting numbers are integer.

Regards,
Max










On Wed, May 16, 2018 at 3:47 PM, Paul Hanna <pauldhanna.math at gmail.com>
wrote:

> 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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>