HMMT 2006 U5

Find the number of solutions in positive integers $\left(k ; a_{1}, a_{2}, \ldots, a_{k} ; b_{1}, b_{2}, \ldots, b_{k}\right)$ to the equation

$$a_{1}\left(b_{1}\right)+a_{2}\left(b_{1}+b_{2}\right)+\cdots+a_{k}\left(b_{1}+b_{2}+\cdots+b_{k}\right)=7 .$$