Have you heard of reciprocal power sum?

Algebra Level 5

$\frac 1 1 + \frac 1 {1^3 + 2^3}+ \frac 1 {1^3 + 2^3 + 3^3 }+ \frac 1 {1^3 + 2^3 + 3^3 +4^3 } + \ldots$

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 $a+b+c$ .