# [seqfan] Re: Displaying an o.g.f for binomial(mn,n)

Thomas Copeland tccopeland at gmail.com
Sat May 26 00:05:11 CEST 2012

Hi Max,

Thanks for your observations, but let me make three points in return:

First, Newton's binomial expansion (and factorial series) is nothing
new.  Aren't we looking for new ways to view the sequences analytically,
geometrically, combinatorally, etc.? (Actually, o.g.f. formulations date
back to the 1730's and de Moivre, so they're not that new.)

Second, the analytic derivation would make the expression obvious and
connect it to an autonomous differential equation, iterated derivatives,
trees, and other things, if correct. I'll write it up soon, since the
expression is certainly true for m=1,2, and 3, (but thoroughly explaining
the expression wasn't my intent in asking for *formatting help*).

Third, the square of the discriminant is explicitly in the equation already
in the denominator out front in the usual notation as delta. The numerator
in the sum is definitely not the discriminant, but is related.

Tom

On Fri, May 25, 2012 at 10:00 PM, Max Alekseyev <maxale at gmail.com> wrote:

> Hi Tom,
>
> This is not about your question but just a side note:
>
> The expression looks to me unnecessarily cryptic. I believe a simpler
> one can be obtained from series multisection
> http://en.wikipedia.org/wiki/Series_multisection#Example of
> (1+x)^{4n}.
> Also the internal product seems to be simply the square root of
> discriminant of the polynomial x^4-x+z w.r.t. x (which can be computed
> even without knowing zeroes of the polynomial) - why not rewrite it
> this way?
>
> Regards,
> Max
>
> On Fri, May 25, 2012 at 3:34 PM, Thomas Copeland <tccopeland at gmail.com>
> wrote:
> > Dear OEIS editors,
> >
> > Could someone please show me how to format the LaTeX expression below
> > acceptably for the OEIS?
> >
> > It represents an apparent o.g.f. (z>=0) for the sequence
> > binomial(4n,n), A005810. (It's integral is related to the associated
> > Fuss-Catalan/Raney sequence A002293.)  I'd like to submit similar
> > expressions for binomial(mn,n) for m=2,3, ....
> >
> > Please paste the following expression in the LaTeX editor at
> > http://www.codecogs.com/latex/eqneditor.php to display it:
> >
> >
> > o.g.f.(z^3)=\frac{1}{2\sqrt\Delta}\sum_{i= 1}^{4}\prod_{j,k\neq
> > i,j<k}^{4}|x_j-x_k|
> >
> >
> > where the x_k are the four zeros for   x^4-x+z=0 and delta is the
> > discriminant |256-27z^3|^(1/2).
> >
> > Sincerely,
> > Tom Copeland
> >
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> >
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