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 squares

Abstract

Let $c_r\left(n\right)$ be the number of representations of a positive integer $n$ as a sum of $r$ squares, where representations with different orders and different signs are counted as distinct. We prove a combinatorial identity relating this quantity to the sum of inverses of odd divisors of $n$:

$\sum _{oddd|n}\frac{1}{d}=\frac{1}{2}\sum _{r=1}^n\frac{{\left(-1\right)}^{n+r}}{r}\left(\genfrac{}{}{0.0pt}{}{n}{r}\right)c_r\left(n\right).$