[seqfan] Re: Help with posting comment for partition numbers

[Olivier Gerard]

On Fri, Aug 22, 2014 at 10:33 AM, Bob Selcoe <rselcoe at entouchonline.net>
wrote:

> Using my own approach and notation is the best I can do at the moment; if
> there is a way to approach these equations in a more conventional and
> meaningful way, I would like to learn.
>
>
That's one of the purposes of the OEIS.
By looking up what you compute with your own notations, you are led
to reference works and classical problems considered by previous
researchers.



> In the meantime, there are a few of your comments  I can address now.
> First, I am somewhat familiar with the pentagonal theorem and understand
> some of how it applies here; clearly, the initial terms of the equations
> involve pentagonals (i.e., MacMahon's recurrence), with variations after.
> As I see it, while the 1+ sum_f(k) equations (for lack of a better term)
> are less elegant than MacMahon's; they offer more variety.  So for P(250),
> MacMahon offers one beautiful equation; the 1+sum_f(k)'s offer 125
> different equations (albeit generally not as beautiful), all related to
> each other and generated by one simple process.    Perhaps that is per se
> interesting, or even potentially useful, and (eventually) worthy of an oeis
> comment?
>
> Yes especially if it leads to new sequences where you can explain the
process in detail and link from them to classical core sequences such as
A000041.
More on this below.


> I had considered that it would be better to create a triangle, based on
> the numeric values in the terms with their attendant coefficients  - I've
> checked and there don't appear to be any corresponding sequences in the
> oeis.  But I don't know the format for including coefficients in triangles,
> so I avoided doing it.  Is there a way to format such a triangle?
>

Yes. Sequence A026794 you have been referring to is a triangle.
The convention is to describe the
triangle as coefficients T(n,k)  (in the same way we use a(n) to refer to
a linear sequence). If your first row is n=1 and your first column is k=1,
you submit values as a sequence in this order:

T(1,1), T(2,1), T(2,2), T(3,1), T(3,2), T(3,3), T(4,1), etc.




>
> re:   >> Referring to the equations f(k) in sequence A026794: a(n) = 1 +
>
>  sum{k=1..k}  f(k), when k+1<=n<=2k+1.
>>>
>>
> Yes, that is not clear.  What I mean is: sum the terms in the equations
> f(1), f(2), f(3)... f(k).  So perhaps sum_{i=1..k} f(i) when  k+1<=n<=2k+1
> makes sense?
>
>
The basic problem is that you have on one side a constraint on i and then a
double constraint on n expressed in term of k.

k+1 <= n <= 2k+1

But k depends on n. Your form is equivalent to this constraint on k:

 (n-1)/2 <= k <= n-1

So what you are trying to discuss is something like

p(n) =  sum_{k = ceiling((n-1)/2) .. (n-1) }( g( n, k) * (sum_{i = 1 ..k}
f(n,k,i) )  )

where my notation provides for the fact that the
fonctions involved (g, f), might depend on n,k and i.
If there is less dependance, you will be able to
avoid quoting some of the variables some of the time.

It is in general a good idea to list relevant variables
instead of being implicit.

g(n,k) would be your triangle of coefficients
and f(n,k,i) would be your extract of the triangle
of partitions by size of the least part.


Olivier