Let m >= 1. The sequence of hypergraph Catalan numbers {C_m(n): n >= 0} is defined in terms of counting walks on trees, weighted by the orders of their automorphism groups. See Gunnells. When m = 1 we get the sequence of Catalan numbers A000108. The present sequence is the case m = 6.
Gunnells gives several combinatorial interpretations of the hypergraph Catalan numbers, a method to compute their generating functions to arbitrary precision and some conjectural asymptotics.
