What is $\binom{n}{-k}$?

Question

What is $\binom{n}{-k}$ ? If $n,k\ge0$

In Wikipedia there's a case where $n$ is negative and not $k$

But if Pascal's rule still holds, I get for example for $k=0$;

$\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{1}$

$1=\binom{n-1}{-1}+1\implies\binom{n}{-1}=0$ for any $n$

Is that true, there must be a generalization of the formula, can you help please.

Answer 1

Many (all, maybe?) results about binomial coefficients hold if you define:

$$\displaystyle \binom n k = 0 \text{ for } k < 0$$