Prove that $\displaystyle \prod_{j=1}^{2n}\left(j^j(2n+j)^{j-1}\right)$ is a perfect square

Prove that $\displaystyle \prod_{j=1}^{2n}\left(j^j(2n+j)^{j-1}\right)$ is a perfect square

For any positive integer $n$, prove that $\displaystyle \prod_{j=1}^{2n}\left(j^j(2n+j)^{j-1}\right)$ is a perfect square.
Let $f(n) = \prod_{j=1}^{2n}\left(j^j(2n+j)^{j-1}\right)$. Then $$f(1) = 2^4,...