证明乘积$\inZ$

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证明$\forall n\ge1,a_n\inZ$$$
a_n=\frac{2^{n^2-1}}{n^2}
\prod_{\substack{0\le m_1,m_2\le n-1\\(m_1,m_2)\neq(0,0)}}
\Bigl(2-\cos{\tfrac{\pi m_1}{n}}-\cos{\tfrac{\pi m_2}{n}}\Bigr)
$$

$n$	$a_n$
1	1
2	4
3	192
4	100 352
5	557 568 000
6	32 565 539 635 200