05A19-GeneratingFunctionForTheReciprocalCentralBinomialCoefficients.tex

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\pmtitle{generating function for  the reciprocal central binomial coefficients}
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It is well known that the sequence called central binomial coefficients is defined by ${2n\choose n}$ and whose initial terms are $1,2,6,20,70, 252,...$ has a generating function $\frac{1}{\sqrt{1-4x}}$.
But it is less known the fact that
the function
$$\frac{4\,\left( {\sqrt{4 - x}} + {\sqrt{x}}\,\arcsin (\frac{{\sqrt{x}}}{2}) \right) }{{\sqrt{(4 - x)^3}}}
$$
has ordinary power series
$$1+\frac{x}{2}+\frac{x^2}{6}+\frac{x^3}{20}+\frac{x^4}{70}+\frac{x^5}{252}+...$$
This means that such a function is a generating function for the reciprocals ${2n\choose n}^{-1}$.

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

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Reference:

1) Renzo Sprugnoli, {\it Sum of reciprocals of the Central Binomial Coefficients}, Integers: electronic 
journal of combinatorial number theory, 6 (2006) A27, 1-18
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