Beautiful Double Sum

Calculus Level 5

\[\sum _{ k=1 }^{ \infty }{ \sum _{ j=1 }^{ \infty }{ \left[ \frac { { H }_{ j }\left( { H }_{ k+1 }-1 \right) }{ kj\left( k+1 \right) \left( j+k \right) } \right] } } =\frac { -A }{ B } { \pi }^{ C }-D\zeta \left( E \right) +\frac { F }{ G } { \pi }^{ H }\zeta \left( I \right) +J\zeta \left( K \right) \]

The above equation is true for positive integers \(A,B, \ldots ,K\), where \(A,B\) and \(F,G\) are pairwise coprime.

Find \(A+B+ \cdots +K\).

Notation: \(\zeta(\cdot) \) denotes the Riemann zeta function.

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