There are 2n points marked on a circle. We want to divide them into pairs and connect the points in each pair by a segment (chord) such that these segments do not intersect.

Let $\displaystyle Q_{n}$ be the number of ways to do this.

Write a recurrence relation for $\displaystyle Q_{n}$.

I worked $\displaystyle Q_{n}$ out up to n=5:

$\displaystyle Q_{0} = 0$

$\displaystyle Q_{1} = 1$

$\displaystyle Q_{2} = 2$

$\displaystyle Q_{3} = 5$

$\displaystyle Q_{4} = 14$

$\displaystyle Q_{5} = 42$

I had worked $\displaystyle Q_{n}$ out to be $\displaystyle 2Q_{n-1} + 2Q_{n-2}$ but this failed when I worked out n=5.