Inverse Sum Quintic

Calculus Level 5

$\large\displaystyle\sum_{n=1}^{\infty} \left({\displaystyle\sum_{k=0}^nk^5} \right)^{-1}$

If the value of the series above can be expressed as $A\pi^2 + \sqrt B \pi \tan \left( \dfrac{\pi \sqrt C}2\right)+ D$ where $$A,B,C$$ and $$D$$ are integers, find the value of $$A+B+C+D$$.

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