KöMaL 2025 A904

Let $n$ be a given positive integer. Luca, the lazy flea sits on one of the vertices of a regular $2n$-gon. For each jump, Luca picks an axis of symmetry of the polygon, and reflects herself on the chosen axis of symmetry. Let $P(n)$ denote the number of different ways Luca can make $2n$ jumps such that she returns to her original position in the end, and does not pick the same axis twice. (It is possible that Luca's jump does not change her position, however, it still counts as a jump.) a) Find the value of $P(n)$ if $n$ is odd. b) Prove that if $n$ is even, then $P(n)=(n-1)!\cdot n!\cdot \sum_{d\mid n,d\in \mathbb{N}}\left(\varphi\left (\dfrac{n}{d}\right) \cdot \binom{2d}{d} \right)$, where $\varphi(k)$ denotes the number of positive integers not greater than $k$ that are coprime to $k$.