Welcome 2016! Part 14

Algebra Level 4

\[ \large \displaystyle \sum_{n=1}^{\infty} \displaystyle \sum_{k=1}^{n-1} \dfrac{k}{2^{n+k}} \]

If the value of the series above is equal to \(\dfrac AB\), where \(A\) and \(B\) are coprime positive integers, find the value of \(B^2-A^2\).

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