A cute find closed form of sum problem

Originally published at 狗和留美者不得入内. You can comment here or there.

A friend pinged me this on Facebook. I decided to look at it to exercise my technical chops. Well, the value of the denominator is given by the hint. In the sum of the first triangular numbers, is summed times, and the number of ways to split items in a line and pick one on each side of the split is the same as the number of ways to select items from , with the middle one representing the split point. Finally do a partial fractions to telescope. You’ll get .