It is simpler than it looks!

Calculus

$\sum _{ n=1 }^{ \infty }{ \frac { 1 }{ { n }^{ 2 }\left( \begin{matrix} 2n \\ n \end{matrix} \right) } } =\frac { A }{ B } \zeta \left( C \right)$

In the equation above, $A,B,C$ are positive integers such that $A$ and $B$ are coprime.

Find $A+B+C.$