In this question, a proof using real analysis is given of the following identity $$ \sum_{n=1}^{\infty} \frac{(n-1)!}{n \prod_{i=1}^{n} (a+i)} = \sum_{k=1}^{\infty} \frac{1}{(a+k)^{2}}$$

Is there a combinatorial proof of this identity? If so, does the proof require that $a$ be a natural number?