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.