Warming up summation

Calculus Level 5

\[\large \sum _{ n=1 }^{ \infty }{ \sum _{ k=n+1 }^{ \infty }{ \dfrac { { H }_{ n } }{ { k }^{ 6 } } } } =\dfrac { { \pi }^{ A } }{ B } -\dfrac { ({ \zeta }( C ))^2 }{ D } -\zeta ( E ) \]

The above equation holds true for positive integers \(A\), \(B\), \(C\), \(D\), and \(E\) with \(C\) and \(E\) are odd numbers.

Find \(A+B+C+D+E\).

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