KöMaL 2016 A661

Let $K$ be a fixed positive integer. Let $(a_0,a_1,\dots)$ be the sequence of real numbers that satisfies $a_0=-1$ and $\sum_{\substack{i_0,i_1,\dots,i_K\ge0 \\ i_0+i_1+\dots+i_K=n}} \frac{a_{i_1}\cdot\dots\cdot a_{i_K}}{i_0+1} =0$ for every positive integer $n$ . Show that $a_n>0$ for $n\ge1$.