Minors of a class of Riordan arrays related to weighted partial Motzkin paths

Some New Binomial Sums Related to the Catalan Triangle

“…Many properties of the Catalan numbers can be generalized easily to the ballot numbers, which have been studied intensively by Gessel [25]. The combinatorial interpretations of the ballot numbers can be found in [5,8,10,11,14,18,19,21,22,28,31,35,39,41,44,50,51]. It was shown by Ma [33] that the Catalan triangle C can be generated by context-free grammars in three variables.…”

Combinatorial identities related to 2 × 2 submatrices of recursive matrices

“…Substitute f (v) into the right-hand side of (3.1) to complete the proof of (3.4). ✷ Note that the above lemma could also be deduced by using the partial weighted Motzkin paths, as was done in [24]. Now we can determine the entries of A (z+1,z+1,z+1,...),(z,z,z,...) .…”

“…It is well known that each entry A n,k admits a weighted partial Motzkin path interpretation, see [1,10,16]. Using the weighted partial Motzkin paths, Sun and Ma [24] obtained the following general result. Theorem 1.1.…”

“…Shapiro's Catalan triangle, defined by B = (B n,k ) n≥k≥0 with B n,k = k+1 n+1 2n+2 n−k [20, A039598], appears in various combinatorial settings [2,17,18,19] and has been paid a lot attention in combinatorics and number theory [3,8,11,12,14,15,23,24,25]. Define another infinite lower triangle X = (X n,k ) n≥k≥0 , based on the triangle B as follows, X n,k = det B n,k B n,k+1 B n+1,k B n+1,k+1 .…”

“…When the parameters (x, y) are specialized, Theorems 1.1 and 1.2 can produce many combinatorial identities related to Catalan numbers, Motzkin numbers and other combinatorial sequences [23,24]. Despite this, there still have many classical combinatorial triangles do not fall into the above framework.…”