For $ n\in\mathbb{Z}_{\ge 1}$ , let $ A_n$ be the $ n\times n$ matrix given by $ (A_n)_{i,j}={n\choose |i-j|}$ . From this post it is clear that $ $ \det(A_n)=\prod_{k=0}^{n-1}\left[\left(\exp\left(\frac{2\pi k i}{n}\right)+1\right)^n-1\right]=\prod_{k=0}^{n-1}\left(2^n(-1)^k\cos^n\left(\frac{\pi k}n\right)-1\right).$ $ Also, $ \det(A_n)$ is obviously integer for all $ n$ .

**Question:** What is known about the (prime) divisors of $ \det(A_n)$ ?

If this is too broad, I am particularly interested in pairs $ (p,d)$ where $ p$ is prime with $ d\mid p-1$ and $ p\nmid\det(A_{(p-1)/d})$ .