generating function for the reciprocal central binomial coefficients

It is well known that the sequence called central binomial coefficients is defined by $\left(\genfrac{}{}{0pt}{}{2n}{n}\right)$ and whose initial terms are $1,2,6,20,70,252,\mathrm{\dots }$ has a generating function $\frac{1}{\sqrt{1-4x}}$.But it is less known the fact thatthe function

$$\frac{4\left(\sqrt{4-x}+\sqrt{x}\arcsin (\frac{\sqrt{x}}{2})\right)}{\sqrt{{(4-x)}^3}}$$

has ordinary power series

$$1+\frac{x}{2}+\frac{x^2}{6}+\frac{x^3}{20}+\frac{x^4}{70}+\frac{x^5}{252}+\mathrm{\dots }$$

This means that such a function is a generating function for the reciprocals ${\left(\genfrac{}{}{0pt}{}{2n}{n}\right)}^{-1}$.

From that expression we can see that the numerical series ${\sum }_{n=0}^{\mathrm{\infty }}{\left(\genfrac{}{}{0pt}{}{2n}{n}\right)}^{-1}$ sums $\frac{4\left(\sqrt{3}+\frac{\pi }{6}\right)}{3\sqrt{3}}$ which has the approximate value $1,7363998587187151$.

Reference:

1) Renzo Sprugnoli, Sum of reciprocals of the Central Binomial Coefficients, Integers: electronicjournal of combinatorial number theory, 6 (2006) A27, 1-18