Harmonic sum with reciprocal central binomial coefficient

Let \mathcal{H}_n denote the n – th harmonic number. Prove that

    \[\sum_{n=1}^{\infty} \frac{2^n \mathcal{H}_n}{\binom{2n}{n}} = \pi - \frac{\pi \log 2}{2} + \mathcal{G}\]

where \mathcal{G} denotes the Catalan’s constant.

Solution

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