I've been trying to complete a problem recently, and have been bashing my head against this particular product $$Q(p)=\prod_{k=1}^{p-1}k^{2k-p-1}$$
The problem states that for any prime $p$, $Q(p)\ \epsilon\ \mathbb{N}$
And while the "$-p-1$" can be simplified into $\frac{1}{((p-1)!)^{p+1}}$, I can't at all make progress with the part that is $k^{2k}$. All it gives me is this numerator: $$1^2\cdot 2^4\cdot 3^6\cdot 4^8\cdots (p-1)^{2(p-1)}$$
Is there any nice way to simplify this expression? My previous attempts yielded far-too-convenient answers that turned out to be wrong. They all really considered the prime factors of each and trying to leverage the fact that $\nexists s \in \{2\ldots (p-1)\}\quad s \mid p$.
I do not think there is a simple closed form. However, you can express it using the $K$-function or the Barnes $G$-function ($+$ the gamma function): $$ \prod\limits_{k = 1}^{p - 1} {k^{2k - p - 1} } = \frac{1}{{(\Gamma (p))^{p + 1} }}\left( {\prod\limits_{k = 1}^{p - 1} {k^k } } \right)^2 = \frac{{K^2 (p)}}{{(\Gamma (p))^{p + 1} }} = \frac{{(\Gamma (p))^{p - 3} }}{{G^2 (p)}} = \frac{{(\Gamma (p))^{p - 1} }}{{G^2 (p + 1)}}. $$