Even harmonic?

Calculus Level 5

\[ \large \sum_{n=1}^\infty (-1)^{n-1} \dfrac{H_{2n}}n = \dfrac{A\pi^B}C - \dfrac{(\ln B)^2}D \]

If the equation above holds true for positive integers \(A,B,C\) and \(D\) with \(A,C\) coprime, find \(A+B+C+D\).

\[\] Notation: \( H_n\) denotes the \(n^\text{th} \) harmonic number, \( H_n = 1 + \dfrac12 + \dfrac13 + \cdots + \dfrac1n\).

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