# 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^{} |
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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 |