05A15-WedderburnEtheringtonNumber.tex

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\pmtitle{Wedderburn-Etherington number}
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\pmsynonym{Wedderburn Etherington number}{WedderburnEtheringtonNumber}
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The $n$th {\em 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.
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