A cute find closed form of sum problem


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 n triangular numbers, k is summed n+1-k times, and the number of ways to split n+1 items in a line and pick one on each side of the split is the same as the number of ways to select 3 items from n+2, with the middle one representing the split point. Finally do a partial fractions to telescope. You’ll get \frac{1/2}{n} - \frac{1}{n+1} + \frac{1/2}{n+2}.