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

[Max Alekseyev]

P.S. B_n is the complete Bell polynomial
https://en.wikipedia.org/wiki/Bell_polynomials

On Sun, May 20, 2018 at 11:46 AM, Max Alekseyev <maxale at gmail.com> wrote:

> 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)!*A30
> 4312(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
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
>