Negative Binomial Coefficient
Definition
For $r,k \in \mathbb{N}$, $\displaystyle \binom{-r}{k}$ is called the Negative Binomial Coefficient.
Explanation
As the name suggests, the negative binomial coefficient is an extension of the binomial coefficient to negative numbers. Mathematically, there’s no reason not to compute it for $\alpha \in \mathbb{Z}$ as shown in $\displaystyle \binom{\alpha}{k} = {{ \alpha ( \alpha - 1 ) \cdots ( \alpha - k + 1 ) } \over { k! }}$.
Furthermore, it can be generalized to complex numbers. In particular, the discussion on negative integers $-r$ has its own unique application, thus earning a separate name. The negative binomial coefficient can also be expressed simply as a binomial coefficient. $$ \begin{align*} \binom{-r}{k} =& {{ (-r) ( -r - 1 ) \cdots ( -r - k + 1 ) } \over { k! }} \\ =& (-1)^{k} {{ r ( r + 1 ) \cdots ( r + k - 1 ) } \over { k! }} \\ =& (-1)^{k} \binom{r + k - 1}{ k } \end{align*} $$ Multiplying both sides by $(-1)^{k}$ gives: $$ (-1)^{k} \binom{-r}{k} = \binom{r + k - 1}{ k } $$ Indeed, in the probability mass function of the negative binomial distribution, such an expression is used.