I have a question regarding this answer. At one point, Alex Francisco writes at the end that $$\prod\limits_{\substack{0 \le \ell \le j \\ \ell \neq k}} ((k + 1)^2 - (\ell + 1)^2) = \dfrac{(-1)^{j - k}}{2(k + 1)^2} \cdot (j - k)!\, (j + k + 2)!$$
How did Alex Francisco do that? And also why is there a 2 in the bottom? It seems as it disappears in the following step so maybe it's a typo?
I tried to expand and I didn't manage to get the same result, and I didn't find where the factor $2$ came from.
I will assume $j \geq k$. $$\prod\limits_{\substack{0 \le \ell \le j \\ \ell \neq k}} ((k+1)^2 -(l+1)^2) = \prod\limits_{\substack{0 \le \ell \le j \\ \ell \neq k}} (k-l)(k+l+2) = \frac{k! \cdot \prod\limits_{l=k+1}^j (k-l) \cdot \prod\limits_{l=0}^j (k+l+2)}{k+k+2} = \frac{(k+j+2)! \cdot (-1)^{j-k} \cdot (j-k)!}{2(k+1)^2} $$