Have you heard of reciprocal power sum?

Algebra Level 4

\[ \frac 1 1 + \frac 1 {1^3 + 2^3}+ \frac 1 {1^3 + 2^3 + 3^3 }+ \frac 1 {1^3 + 2^3 + 3^3 +4^3 } + \ldots \]

If the series above can be expressed as \( \frac {a \pi^2}{b} - c \) for positive integers \(a,b,c\) with \(a,b\) coprime. Find \(a+b+c\).