Intriguing Sum

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

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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