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

TL;DR

This paper establishes a new identity connecting the sum over divisors of a positive integer with the number of its representations as a sum of triangular numbers, using combinatorial and number-theoretic techniques.

Contribution

It introduces a novel identity linking divisor sums and representations as sums of triangular numbers, expanding understanding of their interplay.

Findings

Proves the identity relating divisor sums to representations as triangular numbers

Utilizes a result by Ono, Robbins, and Wahl to establish the identity

Provides a new perspective on the structure of sums of 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.