Raney numbers, threshold sequences and Motzkin-like paths

We provide new interpretations for a subset of Raney numbers, involving threshold sequences and Motzkin-like paths with long up down steps. Given three integers n,k,ℓ such that n≥1,k≥2 0≤ℓ≤k−2, (k,ℓ)-threshold sequence length n is any strictly increasing S=(s1s2…sn) ki≤si≤kn+ℓ. These are in bijection the (ℓ+1)-tuples k-ary trees total internal nodes. prove this result using counting arguments but we also an explicit between tuples trees. further show how to represent as steps, deduce these enumerated by same numbers. Finally, illustrate use finding combinatorial identities.