Why does this work? I think the best is to cram the recursion relationship just like some mathematical formulae. Not sure, I will be able to remember this relation for long. Because my mind would always ask how this relation could be correct. And, I would have to fool my brain. As I can have nothing else to convince it. I really wish if someone can make me understand it or show me how it brings the right output.
I hope this image will help you in understanding the recurrence relation
Explanation: For any A you have 2A points, imagine those points on a circle and pick one point, now draw all chords from this point such that for each chord number of points on both sides of the chord is even. For example when A=3, I have drawn ab, ad and af. Now fixing each chord you need to find number of chords possible on both side of fixed chord. For example when A=4 and fixed chord is ad then I need to find number of chords possible with points [b,c] and [h,g,f,e].