Školjka

%V0 Let $a_{0}$, $a_{1}$, $a_{2}$, $...$ be a sequence of reals such that $a_{0} = - 1$ and $a_{n} + \frac {a_{n - 1}}{2} + \frac {a_{n - 2}}{3} + ... + \frac {a_{1}}{n} + \frac {a_{0}}{n + 1} = 0$ for all $n\geq 1$. Show that $a_{n} > 0$ for all $n\geq 1$.