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We provide new interpretations for a subset of Raney numbers, involvingthreshold sequences and Motzkin-like paths with long up and down steps.  Given three integers n, k, l such that n >= 1, k >= 2 and 0 <= l <= k-2, a(k,l)-threshold sequence of length n is any strictly increasing sequence S=(s_1s_2 ... s_n) of integers such that ki <= s_i <= kn+l. These sequences are inbijection with ordered (l+1)-tuples of k-ary trees. We prove this result andidentify the Raney numbers that count the (k,l)-threshold sequences. As aconsequence, when k=2 and k=3, we deduce combinatorial identities involvingCatalan numbers and powers of 2, and respectively Fuss-Catalan and Raneynumbers. Finally, we show how to represent threshold sequences as Motzkin-likepaths with long up and down steps, and deduce that these paths are enumeratedby the same Raney numbers.

Raney numbers, threshold sequences and Motzkin-like paths