Linear chord diagrams with long chords

A linear chord diagram of size $n$ is a partition of the set $\{1,2,\cdots,2n\}$ into sets of size two, called chords. From a table showing the number of linear chord diagrams of degree $n$ such that every chord has length at least $k$, we observe that if we proceed far enough along the diagonals, they are given by a geometric sequence. We prove that this holds for all diagonals, and identify when the effect starts.