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 $n$ 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 $\frac{1/2}{n} - \frac{1}{n+1} + \frac{1/2}{n+2}$.

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