Harmonic numbers and sums

Calculus Level 5

\[ \displaystyle \sum _{ n=1 }^{ \infty }{ { \left(\frac { { H }_{ n } }{ n } \right) }^{ 2 } } \]

If the above summation can be stated in the form \( \dfrac { A{ \pi }^{ B } }{ C } \) for positive integers \(A,B,C\) with \(\text{gcd}(A,C) = 1\).

What is the value of \(A+B+C\)?

Details and Assumptions

\(H_n \) is the \(n^\text{th} \) Harmonic number, \( \displaystyle {H}_{n} = \sum _{ r=1 }^{ n }{ \frac { 1 }{ r } } \)

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