\[\text{S} = \displaystyle \sum_{n=0}^\infty \left[\binom{2n}{n}^2\frac{ H_n}{2^{5n}} \right] \]

\(\text{S}\) can be represented as

\[\dfrac{\sqrt{\pi^{\text{A}}}}{\text{B} \ \Gamma^2\left(\frac{\text{C}}{\text{D}} \right)}(\text{E}\pi - \text{F}\log \text{G})\]

where \(\text{A}, \text{B}, \text{C}, \text{D}, \text{E}, \text{F}\) and \(\text{G}\) are positive integers, \(\gcd (\text{C},\text{D}) = \gcd(\text{B},\text{E}) = 1\) and \(\text{G}\) is a prime number.

Evaluate \(\text{A}+\text{B}+\text{C}+\text{D}+\text{E}+\text{F}+\text{G}\)

See Also : Part 1

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