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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