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

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