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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\...
T. Amdeberhan's user avatar