Series involving central binomial coefficient squared

In this blog entry we will evaluate the beautiful series below involving the square of the central binomial coefficient

$\begin{align*} \sum_{n=1}^\infty \frac{\binom{2n}{n}^2}{ 2^{4n} n}&=4 \ln(2)-\frac{8G}{\pi} \end{align*}$

Click here to see the proof

We used the previous result

$\begin{align*} \sum_{n=1}^\infty \binom{2n}{n}\frac{x^n}{ 2^{2n} n}&=2\ln\left(\frac{2}{1+\sqrt{1-x}} \right) \end{align*}$

proved here

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