you lost me with the edit.

do you mean this?

$z=\dfrac{\left(\begin{array}{c}y \\ \frac{x+y}{2}\end{array}\right)}{\displaystyle{ \sum_{x=0}^y} \left(\begin{array}{c}y \\ x\end{array}\right)}$

and you want to show $z=0$ ? It's pretty clearly not 0 as all those terms are positive.