# generating function for the reciprocal Catalan numbers

The series

$$1+x+\frac{{x}^{2}}{2}+\frac{{x}^{3}}{5}+\frac{{x}^{4}}{14}+\frac{{x}^{5}}{42}+\frac{{x}^{6}}{132}+\frac{{x}^{7}}{429}+\mathrm{\cdots}$$ |

whose coefficients are the reciprocal of the Catalan numbers^{} $\frac{\left(\genfrac{}{}{0pt}{}{2n}{n}\right)}{n+1}$, has as a generating function

$$\frac{2\left(\sqrt{4-x}\left(8+x\right)+12\sqrt{x}\mathrm{arctan}(\frac{\sqrt{x}}{\sqrt{4-x}})\right)}{\sqrt{{\left(4-x\right)}^{5}}}$$ |

To deduce such a formula the easy way, one starts from the generating function of the reciprocal central binomial coefficients^{} and having into account the relation

$$\frac{d}{dx}\left(\frac{{x}^{n+1}}{\left(\genfrac{}{}{0pt}{}{2n}{n}\right)}\right)=\frac{(n+1){x}^{n}}{\left(\genfrac{}{}{0pt}{}{2n}{n}\right)}$$ |

for each term in the corresponding series and applied to the function^{} in the region of uniform convergence^{}.
Another method is almost exaclty the same like in the derivation of the generating function for the reciprocal central binomial coefficients.

Title | generating function for the reciprocal Catalan numbers |
---|---|

Canonical name | GeneratingFunctionForTheReciprocalCatalanNumbers |

Date of creation | 2013-03-22 19:05:12 |

Last modified on | 2013-03-22 19:05:12 |

Owner | juanman (12619) |

Last modified by | juanman (12619) |

Numerical id | 10 |

Author | juanman (12619) |

Entry type | Derivation |

Classification | msc 05A19 |

Classification | msc 05A15 |

Classification | msc 05A10 |

Related topic | CatalanNumbers |