PL-OM 2007 Problem 6

Sequence $a_{0}, a_{1}, a_{2},...$ is determined by $a_{0}=-1$ and $a_{n}+\frac{a_{n-1}}{2}+\frac{a_{n-2}}{3}+...+\frac{a_{1}}{n}+\frac{a_{0}}{n+1}=0$ for $n\geq 1$ Prove that $a_{n}>0$ for $n\geq 1$