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 $$\sum _{d|n}\frac{1+2(-1{)}^d}{d}=\sum _{r=1}^n\frac{(-1{)}^r}{r}\left(\genfrac{}{}{0ex}{}{n}{r}\right)t_r(n)$$ using a result of Ono, Robbins and Wahl.