# generating function for the reciprocal alternating central binomial coefficients

It is also not very well known this relation:

 $\frac{4\,\left(\,{\sqrt{x+4}}-{\sqrt{x}}\,{\rm{arcsinh}}(\frac{{\sqrt{x}}}{2})% \right)}{\sqrt{(x+4)^{3}}}=1-\frac{x}{2}+\frac{x^{2}}{6}-\frac{x^{3}}{20}+% \frac{x^{4}}{70}-\frac{x^{5}}{252}+\frac{x^{6}}{924}-...$

where one clearly appreciate that the function on LHS generates the sequence $(-1)^{n}{2n\choose n}^{-1}$.

To obtain the relation above one should use some kind of software because for the function is “terrible” to calculate derivatives of any order. It is a little challenge to give a recursive formula that gives the inverses of these alternating central binomial numbers, when evaluated at $x=0$ at those derivatives.

Title generating function for the reciprocal alternating central binomial coefficients GeneratingFunctionForTheReciprocalAlternatingCentralBinomialCoefficients 2013-03-22 18:58:12 2013-03-22 18:58:12 juanman (12619) juanman (12619) 14 juanman (12619) Example msc 32A05 msc 11B65 msc 05A19 msc 05A15 msc 05A10