$\large \sum_{p=1}^{15} \frac {p \cdot 2^p}{(p+2)!} = w - \frac {x^y}{z!}$

The equation above holds true for some positive integers $w$, $x$, $y$, and $z$. Evaluate $\begin{vmatrix} w & x \\ y & z \end{vmatrix}$.

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