The Combinatorics of M' = 3M~3T and T’ = T~3M

摘要

The central binomial coefficients have the generating function B := ∑(_n~(2n))z~n = 1/(1-4z)~1/2 and it is easily shown that B' = 2B~3. The ternary analog is the generating function M := ∑(_n~(3n)z~n and the situation is more interesting. The analogous identity is M' = 2M~3T where T = ∑1/2n+1(_n~(3n)) z~n give a combinatorial proof by interpreting M,T, and M' in terms of even trees which are ordered trees where the outdegree of every vertex is even.