<s>[INST] 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 $$f(1) = 2^4, \quad f(2) = 2^{20} \cdot 3^4 \cdot 7^2, \quad f(3) = 2^{32} \cdot 3^{18} \cdot 5^8 \cdot 11^4.$$ I didn't see a way to prove the expression is a perfect square for all $n$.
 [/INST] Extracting even powers, we see that, up to a perfect square, this expression is equal to
$$
\prod_{\text{odd }j\le 2n} j \cdot \prod_{\text{even }j\le 2n}(2n+j) = (2n-1)!! \cdot \prod_{k=1}^n(2n+2k) = (2n-1)!! \cdot 2^n \frac{(2n)!}{n!} \\
= (2n-1)!! \cdot 2^{2n} \frac{(2n)!}{(2n)!!} = \bigl(2^n (2n-1)!!\bigr)^2.
$$
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