Product of squared sines: $\prod_{k=1}^{n-1}\prod_{j=1}^{n-1}\left[ \sin^2\left(\frac{k\pi}{2 n}\right) +\sin^2\left(\frac{j\pi}{2 n}\right)\right]$

Asked Apr 13 '17 at 14:52

Active Apr 14 '17 at 07:38

Viewed 235 times

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.

edited Apr 14 '17 at 07:38

Ng Chung Tak

asked Apr 13 '17 at 14:52

Arne Decadt

2

Looks a lot like domino tiling formula, so perhaps backtracking to a eigenvalue problem would help. – Phicar Apr 13 '17 at 15:11
1

I don't get integers: e.g. $a(3) = 3675/2$. – Robert Israel Apr 13 '17 at 15:19
@Phicar I will look into that. Thank you! – Arne Decadt Apr 13 '17 at 15:35
1

@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. – Arne Decadt Apr 13 '17 at 15:35
5

Appears to be http://oeis.org/A007341 . The OEIS entry does not give any closed form in the Formula section, but it does give two alternatives. – Matthew Conroy Apr 13 '17 at 19:44

0 Answers0