# generating function for the reciprocal central binomial coefficients

It is well known that the sequence^{} called central binomial coefficients^{} is defined by $\left(\genfrac{}{}{0pt}{}{2n}{n}\right)$ and whose initial terms are $1,2,6,20,70,252,\mathrm{\dots}$ has a generating function $\frac{1}{\sqrt{1-4x}}$.
But it is less known the fact that
the function

$$\frac{4\left(\sqrt{4-x}+\sqrt{x}\mathrm{arcsin}(\frac{\sqrt{x}}{2})\right)}{\sqrt{{(4-x)}^{3}}}$$ |

has ordinary power series

$$1+\frac{x}{2}+\frac{{x}^{2}}{6}+\frac{{x}^{3}}{20}+\frac{{x}^{4}}{70}+\frac{{x}^{5}}{252}+\mathrm{\dots}$$ |

This means that such a function is a generating function for the reciprocals ${\left(\genfrac{}{}{0pt}{}{2n}{n}\right)}^{-1}$.

From that expression we can see that the numerical series ${\sum}_{n=0}^{\mathrm{\infty}}{\left(\genfrac{}{}{0pt}{}{2n}{n}\right)}^{-1}$ sums $\frac{4\left(\sqrt{3}+\frac{\pi}{6}\right)}{3\sqrt{3}}$ which has the approximate value $1,7363998587187151$.

Reference:

1) Renzo Sprugnoli, Sum of reciprocals of the Central Binomial Coefficients, Integers: electronic journal of combinatorial number theory, 6 (2006) A27, 1-18

Title | generating function for the reciprocal central binomial coefficients |
---|---|

Canonical name | GeneratingFunctionForTheReciprocalCentralBinomialCoefficients |

Date of creation | 2013-03-22 18:58:09 |

Last modified on | 2013-03-22 18:58:09 |

Owner | juanman (12619) |

Last modified by | juanman (12619) |

Numerical id | 12 |

Author | juanman (12619) |

Entry type | Result |

Classification | msc 05A19 |

Classification | msc 11B65 |

Classification | msc 05A10 |

Classification | msc 05A15 |

Synonym | convergent series^{} |