5 we give a proof of our main theorem by using a different recursion which is obtained from a representation theoretical result proved in Section 6.

*Acknowledgements: R.B. thanks Francois Bergeron, Ira Gessel and Dennis Stanton for many fruitful discussions about Chebyshev polynomials which was the starting point of this work. D.K. thanks Christian Krattenthaler for many helpful discussions on the combinatorics of Dyck paths and for pointing him to formula . R.B. also gratefully acknowledges the funding received from NSERC discovery grant of her Postdoc supervisor Michael Lau and Université Laval for the hospitality.*

Excellent filtrations and graded multiplicities {#section2}
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In this section we recall the notion of excellent filtrations (Demazure flags) [@Jo03; @J06] and set up the notation needed in the rest of the paper.

Throughout this paper we denote by $\mathbb{C}$ the field of complex numbers and by $\mathbb{Z}$ (resp. $\mathbb{Z}_{+}$, $\mathbb{N}$) the subset of integers (resp. non-negative, positive integers). Given $n,m\in{\mathbb Z}$, set $$\begin{gathered}
{\genfrac[]{0pt}0{n}{m}}_q=\frac{(1-q^n)\cdots(1-q^{n-m+1})}{(1-q)\cdots (1-q^m)}, \ \ \  n\geq m>0 ,\\  \\ {\genfrac[]{0pt}0{n}{0}}_q=1,\ \  n\geq 0,\ \ {\genfrac[]{0pt}0{n}{m}}_q= 0,\ \  \text{otherwise}.\end{gathered}$$ Further, we introduce the $q$–Pochammer symbol $(a;q)_{n}=\prod_{i=0}^{n-1}(1-aq^{i}).$

Let ${\mathfrak{sl}}_2[t]\cong{\mathfrak{sl}}_2\otimes {\mathbb C}[t]$ the Lie algebra of two by two matrices of trace zero with 