Harmonic numbers and sums

Calculus Level 5

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

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

What is the value of A+B+CA+B+C?

Details and Assumptions

HnH_n is the nthn^\text{th} Harmonic number, Hn=r=1n1r \displaystyle {H}_{n} = \sum _{ r=1 }^{ n }{ \frac { 1 }{ r } }

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