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Calculus Level 5

\[\large \sum _{ k=1 }^{ \infty }{ { \left( 2\sum _{ n=1 }^{ \infty }{ \dfrac { 1 }{ { n }^{ 2 }+kn } } \right) }^{ 2 } } =\dfrac { A }{ B } { \pi }^{ C }\]

The equation above holds true for positive integers \(A,B\) and \(C\) with \(A,B\) coprime. Find \(A+B+C\).

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