# Wedderburn-Etherington number

The $n$th 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 WedderburnEtheringtonNumber 2013-03-22 16:49:32 2013-03-22 16:49:32 PrimeFan (13766) PrimeFan (13766) 6 PrimeFan (13766) Definition msc 05A15 Wedderburn Etherington number Etherington-Wedderburn number