ISL 2006 A2

The sequence of real numbers $a_{0}, a_{1}, a_{2}, \ldots$ is defined recursively by $$a_{0}=-1, \sum_{k=0}^{n} \frac{a_{n-k}}{k+1}=0 \text { for } n \geq 1$$ Show that $a_{n}>0$ for $n \geq 1$. (Poland)