I have a double product

$$a(n)=2^{2 (n-1)^2} \prod_{k=1}^{n-1} \prod_{j=1}^{n-1} \left[ \sin^2 \left(\frac{k\pi}{2 n}\right) +\sin^2 \left(\frac{j\pi}{2 n}\right) \right]$$

which always gives integers. I would like to have a closed form for this or something more intuitive without sines/complex numbers.

I have tried methods from similar questions but I can't seem to evaluate this expression.

Every new insight is appreciated. Thank you in advance.

  • 2
    $\begingroup$ Looks a lot like domino tiling formula, so perhaps backtracking to a eigenvalue problem would help. $\endgroup$ – Phicar Apr 13 '17 at 15:11
  • 1
    $\begingroup$ I don't get integers: e.g. $a(3) = 3675/2$. $\endgroup$ – Robert Israel Apr 13 '17 at 15:19
  • $\begingroup$ @Phicar I will look into that. Thank you! $\endgroup$ – Arne Decadt Apr 13 '17 at 15:35
  • 1
    $\begingroup$ @RobertIsrael You are right. I accidentally wrote it to n instead of to n-1, $a(3)=192$. I changed that now. Thank you for the fast reaction. $\endgroup$ – Arne Decadt Apr 13 '17 at 15:35
  • 4
    $\begingroup$ Appears to be oeis.org/A007341 . The OEIS entry does not give any closed form in the Formula section, but it does give two alternatives. $\endgroup$ – Matthew Conroy Apr 13 '17 at 19:44

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