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The discussion on this page is motivated by Johann Cigler's MO question. My intention arose from a possible generalization of Cigler's matrix $$A_{n,m}=\left( \binom{2m}{j-i+m}-\binom{2m}{m-i-j-1} \...

Let ${C_n}(q)$ be the weight of the Dyck paths of semilength $n$ where the upsteps have weight $1$ and the downsteps which end on height $i$ have weight $q^i$. They satisfy ${C_n}(q) = \sum\limits_{j ...