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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