# Has anyone seen these binary trees (Catalan-type related to the Gegenbauer polynomials and Motzkin paths)?

The OEIS entry A121448 enumerates binary trees with $n$ edges and $k$ vertices with outdegree 1.

Has anyone seen these trees?

The o.g.f. for this entry, $G(x,t)$, is essentially a discriminant whose associated row polynomials can be generated as an umbral composition of the reverse face polynomials of the hypercubes A038207 with the aerated Catalan numbers A126120. The compositional inverse of $G$ is the o.g.f. of the Chebyshev polynomials of the second kind A053117 (mod signs). More generally, the row polynomials of G are invariant polynomials of the compositional inverses of the o.g.f.s (times a factor $x^{\alpha}$) of the Gegenbauer polynomials of differing order $\alpha$. The row polynomials of $G$ are also related to the Motzkin paths of A097610.

Given the rich associations among these combinatoric structures, polygon dissections, lattice paths, orthogonal polynomials, Cartan-Lie algebras, and compositional inverses and algebraic geometry/topology, I would hope these binary trees (or row polynomials) have explicitly popped up in other literature.

• Added the statistic to FindStat at findstat.org/St000385. But no relations to other statistics came up :-( – Christian Stump Feb 10 '16 at 9:02
• @Christian, thanks. Of course, the non-aerated, reflected array has shown up in several publications, but I'm hoping for a little more. – Tom Copeland Feb 10 '16 at 10:56