Negative binomial coefficient
There are $k$ terms which need to be multiplied by $(-1)$ to get the desired quantity. So actually, factoring out the negatives would lead to $(-1)^{2k} = 1$ for all $k$ instead of $(-1)^{k+1}$.
There are $k$ terms which need to be multiplied by $(-1)$ to get the desired quantity. So actually, factoring out the negatives would lead to $(-1)^{2k} = 1$ for all $k$ instead of $(-1)^{k+1}$.