Inverse Sum Cubed

Calculus Level 4

\[\large\displaystyle\sum_{n=1}^{\infty} \left({\displaystyle\sum_{k=0}^nk^3} \right)^{-1} \]

If the value of the series above can be expressed as \(\dfrac{a\pi^2}b-c\) where \(a,b,c\) are positive integers, and \(a,b\) are coprime to each other, find the value of \(a+b+c\).