Wedderburn-Etherington number
The Wedderburn–Etherington numbers are an integer sequence named for Ivor Malcolm Haddon Etherington and Joseph Wedderburn that can be used to count certain kinds of binary trees. The first few numbers in the sequence are
These numbers can be used to solve several problems in combinatorial enumeration. The nth number in the sequence (starting with the number 0 for n = 0) counts
The Wedderburn–Etherington numbers may be calculated using the recurrence relation
beginning with the base case .
In terms of the interpretation of these numbers as counting rooted binary trees with n leaves, the summation in the recurrence counts the different ways of partitioning these leaves into two subsets, and of forming a subtree having each subset as its leaves. The formula for even values of n is slightly more complicated than the formula for odd values in order to avoid double counting trees with the same number of leaves in both subtrees.
The Wedderburn–Etherington numbers grow asymptotically as
where B is the generating function of the numbers and ρ is its radius of convergence, approximately 0.4027 (sequence in the OEIS), and where the constant given by the part of the expression in the square root is approximately 0.3188 (sequence in the OEIS).
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