Intriguing Sum

Discrete Mathematics Level 5

$S=\sum_{n=0}^{\infty}\sum_{k=0}^n2^{-n}(-1)^k{n \choose k}(k+1)^{-4}$

Write $$S=\dfrac{a\pi^c}{b}$$, where $$a,b$$ and $$c$$ are positive integers and $$a,b$$ are coprime. Enter $$a+b+c$$ as your answer.

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