An identity involving number of representations of $n$ as a sum of $r$ triangular numbers

Abstract

Let $\sum_{d|n}$ denote sum over divisors of a positive integer $n$, and $t_{r}(n)$ denote the number of representations of $n$ as a sum of $r$ triangular numbers. Then we prove that $$ \sum_{d|n}\frac{1+2\,(-1)^{d}}{d}=\sum_{r=1}^{n}\frac{(-1)^{r}}{r}\, \binom{n}{r}\, t_{r}(n) $$ using a result of Ono, Robbins and Wahl.