For any integer $n\geq1$, we consider a set $P_{2n}$ of $2n$ points placed equidistantly on a circle. A perfect matching on this point set is comprised of $n$ (straight-line) segments whose endpoints constitute $P_{2n}$. Let $\mathcal{M}_{n}$ denote the set of all non-crossing perfect matchings on $P_{2n}$. A perfect matching $M\in \mathcal{M}_{n}$ is said to be centrally symmetric, if it is invariant under point reflection at the circle center. Determine, as a function of $n$, the number of centrally symmetric perfect matchings within $\mathcal{M}_{n}$. Proposed by Mirko Petrusevski