# generating function for the reciprocal central binomial coefficients

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

 $1+\frac{x}{2}+\frac{x^{2}}{6}+\frac{x^{3}}{20}+\frac{x^{4}}{70}+\frac{x^{5}}{2% 52}+...$

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 $1,\!7363998587187151$.

Reference:

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

Title generating function for the reciprocal central binomial coefficients GeneratingFunctionForTheReciprocalCentralBinomialCoefficients 2013-03-22 18:58:09 2013-03-22 18:58:09 juanman (12619) juanman (12619) 12 juanman (12619) Result msc 05A19 msc 11B65 msc 05A10 msc 05A15 convergent series   