[seqfan] Re: Natural log e in the Pascal system

[Olivier Gerard]

On Sat, Sep 7, 2013 at 7:45 PM, Jess Tauber <yahganlang at gmail.com> wrote:

> Nice site, Peter, but I don't know whether G+ would make me take Google
> Chrome, which I hate.
>
> No. You need a recent browser to use it conveniently but Chrome is not
required.


> In any case I wanted to see whether Harlan Brothers' procedure would give
> interesting results on other straight-line relations in the Pascal system.
> The only other finite sequences, over positive powers of ten, come from the
> shallow diagonals.
>
> Positive powers of ten are irrelevant to the Binomial triangle.


> If you take these terms instead of those from rows you get TWO alternating
> patterns. In the first the resulting numbers keep getting smaller, while
> those of the second keep getting larger, in a regular way determined by
> multiplying previous results by sequences of fractions. I had hoped they
> might end up as limits the way the row results do, and related to e, but
> they don't.
>
> And the results are set up in such a way that if the two sets are properly
> aligned, products of pairs of terms, one from each set, are always exactly
> 2. Could this coordination imply a relationship to some constant of a
> different identity?
>

It is probably only related to the fact that the sum of binomial
coefficients is 2^n.

Olivier