How could I prove that $$\sum _{m=v}^n \left(\left(\prod _{k=v}^{m-1} \frac{k^2}{m^2-k^2}\right)\left(\prod _{k=m+1}^n \frac{k^2}{k^2-m^2}\right)(-1)^{m-v}\right)=1$$ or, simplified, $$\sum _{m=v}^n \prod _{k=v, k \neq m}^{n} \frac{k^2}{k^2-m^2}=1$$

for any positive integers $v$ and $n$, $v \leq n$? I feel this could be somehow related to binomial coefficient identities.

**Why I want it to be true?**

I got this problem while generating formula for eigenvalues of matrix of special type. I noticed that this $$\sum _{m=u}^n \frac{2(-1)^{m-1}(n!)^2}{m^2(n-m)!(n+m)!} \frac{m (m+u-1)!}{u (2u-1)! (m-u)!}(-4)^{u-1}$$ can be simplified to this $$\frac{2((u-1)!)^24^{u-1}}{(2u)!}$$