Arithmetic properties for a certain family of knot diagrams

In this note, we consider arithmetic properties of the function K(n)= (2n)!(2n + 2)!/(n - 1)!(n + 1)!²(n + 2)! which counts the number of two-legged knot diagrams with one self-intersection and n - 1 tangencies. This function recently arose in a paper by Jacobsen and Zinn-Justin on the enumeration of knots via a transfer matrix approach. Using elementary number theoretic techniques, we prove various results concerning K(n), including the following: K(n) is never odd, K(n) is never a quadratic residue modulo 3, and K(n) is never a quadratic residue modulo 5.