# Wedderburn-Etherington number

The $n$th Wedderburn-Etherington number^{} counts how many weakly binary trees can be constructed such that each graph vertex (not counting the root vertex) is adjacent to no more than three other such vertices, for a given number $n$ of nodes. The first few Wedderburn-Etherington numbers are 1, 1, 1, 2, 3, 6, 11, 23, 46, 98, 207, 451, 983, etc. listed in A001190 of Sloane’s OEIS. Michael Somos gives the following recurrence relations:

$${a}_{2n}=\frac{1}{2}{a}_{n}{a}_{n+1}+\sum _{i=1}^{n}{a}_{i}{a}_{2n-i}$$ |

and

$${a}_{2n-1}=\sum _{i=0}^{n-1}{a}_{i+1}{a}_{2n-i}$$ |

with ${a}_{1}={a}_{2}=1$ in both relations^{}.

Title | Wedderburn-Etherington number |
---|---|

Canonical name | WedderburnEtheringtonNumber |

Date of creation | 2013-03-22 16:49:32 |

Last modified on | 2013-03-22 16:49:32 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 6 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 05A15 |

Synonym | Wedderburn Etherington number |

Synonym | Etherington-Wedderburn number |