On page 12 of the paper [Enumeration of chord diagrams on many intervals and their non-orientable analogs][1]" by Alexeev, Andersen, Penner, and Zograf is a list of polynomials which are a refinement of the first few Motzkin polynomials listed in OEIS [A055151][2]. 

Can anyone provide a proof that this relation between the two sets of polynomials holds for all higher degrees in general?


  [1]: http://arxiv.org/abs/1307.0967
  [2]: https://oeis.org/A055151