A double series with double factorials

Calculus Level 5

\[\large \sum_{a=3}^\infty\sum_{b=0}^\infty\frac{(a-2)(2b+2)!!}{(2b+2a)!!} =\frac12\left(x+y\ e^{z/4}\right)\]

The equation above holds true for some integers \(x,y\) and \(z\), compute \(y+z-x\).

Notation: \(!!\) denotes the double factorial.

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