An identity for the sum of inverses of odd divisors of $n$ in terms of the number of representations of $n$ as a sum of $r$ squares

Abstract

Let $$\sum_{\substack{d|n\\ d\equiv 1 (2)}}\frac{1}{d}$$ denote the sum of inverses of odd divisors of a positive integer $n$, and let $c_{r}(n)$ be the number of representations of $n$ as a sum of $r$ squares where representations with different orders and different signs are counted as distinct. The aim is of this note is to prove the following interesting combinatorial identity: $$ \sum_{\substack{d|n\\ d\equiv 1 (2)}}\frac{1}{d}=\frac{1}{2}\,\sum_{r=1}^{n}\frac{(-1)^{n+r}}{r}\,\binom{n}{r}\, c_{r}(n). $$