generating function for the reciprocal alternating central binomial coefficients

It is also not very well known this relation:

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

where one clearly appreciate that the function on LHS generates the sequence ${(-1)}^n{\left(\genfrac{}{}{0pt}{}{2n}{n}\right)}^{-1}$.

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
Canonical name	GeneratingFunctionForTheReciprocalAlternatingCentralBinomialCoefficients
Date of creation	2013-03-22 18:58:12
Last modified on	2013-03-22 18:58:12
Owner	juanman (12619)
Last modified by	juanman (12619)
Numerical id	14
Author	juanman (12619)
Entry type	Example
Classification	msc 32A05
Classification	msc 11B65
Classification	msc 05A19
Classification	msc 05A15
Classification	msc 05A10