# How large do $r$-dimensional “Kasteleyn-Temperley-Fisher” numbers grow?

I brought up a couple of combinatorial and number-theoretic items with this MO question. Now, I shall inquire on growth estimates. Recall $$K_r(n):=\prod_{\ell_1=1}^n\cdots\prod_{\ell_r=1}^n\left( 4\cos^2\left(\frac{\pi\ell_1}{2n+1}\right)+\cdots+4\cos^2\left(\frac{\pi\ell_r}{2n+1}\right)\right).$$

Question. Fix $r\geq3$. Can you provide an asymptotic (at least of first order) for $K_r(n)$, as $n\rightarrow\infty$?

Example. $K_1(n)=1$. It's also known (K.T.F.) that $K_2(n)\sim e^{\frac{4G}{\pi}n^2}$ where $G=$ Catalan's constant.

• Wasn't this answered here: $\lim_{n\rightarrow\infty}n^{-r}\log K_r(n)=2\log 2+C_r$, with $C_r$ some numerical coefficient given by an integral over the $r$-dimensional unit cube [$C_3=0.287095$, $C_4=0.613413$, $C_5=0.856191$, ...] – Carlo Beenakker Apr 12 '17 at 8:12