# Factorials... This Reminds Me of Something..

Calculus Level 5

The infinite sum

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

can be expressed as $$\displaystyle k\frac{a\sqrt{a}-b}{c}\pi^m$$ , Where $$a, b, c,k$$ 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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