Factorials... This Reminds Me of Something..

Calculus Level 5

n=01(n+1)!(12n)!\large \sum_{n=0}^{\infty} \frac{1}{(n+1)! \left(\frac{1}{2} - n\right)!}

The infinite sum above can be expressed as kaabcπm\displaystyle k\frac{a\sqrt{a}-b}{c}\pi^m, where aa, bb, cc, and kk are positive integers, k k and c c are coprime. Find k+a+b+c+m4.\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!=Γ(x+1) x! = \Gamma(x+1) .

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