Does anyone want to add the general binomial theorem (for all n). Also I have some weird alternative formulas for C(n,r) with n a non-negative integer, but I'd have to set up the pictures on a stable server or something.

Definition of binomial coefficients for n < 0

To Maxal and whoever else is willing to comment:

I would say that the recently inserted equation

is reduntant, because of

which shows up below.

The latter formula makes sense only for integer . While the former one is true for any n. --Maxal 22:47, 13 Apr 2005 (UTC)

The point is, everywhere in this article it is assumed that n ≥ 0. Just read through it. The Pascal triangle for example, does not hold for n < 0. I find your addition not in the right place. I think it rather belongs in a generalization section at the end of the article. What do you think? Oleg Alexandrov 23:03, 13 Apr 2005 (UTC)

Disagree. It's the only complete definition for now. And it should come first. I think Pascal triangle and other middle-school-level stuff should form a separate section. --Maxal 23:07, 13 Apr 2005 (UTC)

Please notice that the generalization to n < 0, actually, for any n complex, is at already in the article, see several sections below in there. Oleg Alexandrov 23:10, 13 Apr 2005 (UTC)

First of all, there is a bad mixture of notations. or C(n,k) is the number of combinations and it's defined only for non-negative integer n,k (Pascal's triangle is actually defined for C(n,k)). The binomial coefficient is a generalization of which is defined for integer k and arbitrary n. As of generalization of to complex n, it's ok except for notation. C(n,k) is inappropriate.

Second, the article makes accent on combinatorics where complex n have no (or very limited) application. Hence, it's ok to keep the generalization to complex n at the end. --Maxal 23:26, 13 Apr 2005 (UTC)

So, what are the applications of negative n to combinatorics? Oleg Alexandrov 23:28, 13 Apr 2005 (UTC)

Say, they often appear in expansions of generating functions. And there many self-reverse relations including them. --Maxal 23:37, 13 Apr 2005 (UTC)

So, if you want to really make some changes to this article, you should read it all, then see how to make things look nice overall. What do you think?

I've read it all, and there was no correct definition for binomial coefficients for n<0. Moreover, in the discussion you can see a request for the definition "general" binomial coefficient. I've provided a general definition of binomial coefficients (most popular in combinatorics). From this very same expansion binomial coefficients can be also defined for non-integer n but that's mainly related to analysis. --Maxal 22:47, 13 Apr 2005 (UTC)

Also, please see the sentence:

This is generalized by the binomial theorem, which allows the exponent n to be negative or a non-integer.

in the article. Did you notice it? Oleg Alexandrov 22:42, 13 Apr 2005 (UTC)

Yes but that article should not prevent for providing correct and complete definition of the binomial coefficients. --Maxal 22:47, 13 Apr 2005 (UTC)