A Complicated Binomial Sum

Calculus Level 5

\[\large \displaystyle \sum_{n=0}^{\infty} \dbinom{4n}{2n} 32^{-n} = \sqrt{A}\cos\left(\dfrac{\pi}{B}\right) \]

If the equation above holds true for positive integers \(A\) and \(B\), then find \(A\times B\).

\[\] Notation: \( \dbinom nk = \dfrac{n!}{k!(n-k)!} \) denotes the binomial coefficient.

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