three binomial(n,k) definitions

[Michael Somos]

seqfan,

Brendan McKay wrote :

> Some consistency in mathematical terminology is important and

and this is certainly true, although not always observed in practice.
Witness the situation in Elliptic function theory. Here there is a
profusion of notation and terminology which is not well standardized.
There are several reasons that that situation and they apply in general
to other cases.

   In the case of binomial(n,k), the clearly related case is that
of n! versus \Gamma(z). The gamma function extends the factorial
function, but even Gauss himself used \Pi(z) in some writing for
\Gamma(1+z) with the reason that \Pi(n)=n! versus \Gamma(1+n)=n!.
There is a combinatorial interpretation of the Pascal triangle
and the notation C(n,k) is sometimes used for this purpose to
represent the number of ways of choosing a k-subset of an n-set.
However, I still think that regardless, there is a need for having
extended versions of C(n,k) or binomial(n,k) and what are you going
to write them as? Also, that is not going to change the fact that
different CAS use different definitions.

   I still think there needs to be a discussion about two-way versus
one-way infinite sequences in OEIS because the issue will keep coming
up again in the future. Shalom, Michael