Consider a convex \((n+2)\)-gon. Let \(a_n\)​ denote the number of ways to add non-crossing chords to this polygon, including the case where no chords are added (i.e., \(a_0=a_1=1\) and \(a_2=3\)).

Find a recurrence relation for the sequence \(a_n​\) and determine its generating function.