OIM 2005 Problem 6

Let $n$ be a fixed positive integer. The points $A_1$ , $A_2$ , $\ldots$ , $A_{2n}$ are on a straight line. Color each point blue or red according to the following procedure: draw $n$ pairwise disjoint circumferences, each with diameter $A_iA_j$ for some $i \neq j$ and such that every point $A_k$ belongs to exactly one circumference. Points in the same circumference must be of the same color.

Determine the number of ways of coloring these $2n$ points when we vary the $n$ circumferences and the distribution of the colors.