$\large \sum_{n=0}^{\infty} \frac{1}{(n+1)! \left(\frac{1}{2} - n\right)!}$

The infinite sum above can be expressed as $\displaystyle k\frac{a\sqrt{a}-b}{c}\pi^m$, where $a$, $b$, $c$, and $k$ are positive integers, $k$ and $c$ are coprime. Find $\displaystyle k+a+b+c+m -4.$

**Note:** Factorials of real numbers that are not non-negative integers, is well defined using the Gamma function. We have $x! = \Gamma(x+1)$.

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