What is wrong with this series?

Calculus Level 5

$\large \sum_{n=1,3,5,\ldots}^\infty \dfrac{k \sin^2 \left( \frac nk \right) } {n^2}$

For positive $$k$$, if the series above equals to $$\dfrac AB \pi^C$$ for positive integers $$A,B$$ and $$C$$ with $$\gcd(A,B) = 1$$, find the value of $$A+B+C$$.

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