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 $$a+b+c$$.

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