  We provide new interpretations for a subset of Raney numbers, involving
threshold 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_1
s_2 ... s_n) of integers such that ki <= s_i <= kn+l. These sequences are in
bijection with ordered (l+1)-tuples of k-ary trees. We prove this result and
identify the Raney numbers that count the (k,l)-threshold sequences. As a
consequence, when k=2 and k=3, we deduce combinatorial identities involving
Catalan numbers and powers of 2, and respectively Fuss-Catalan and Raney
numbers. Finally, we show how to represent threshold sequences as Motzkin-like
paths with long up and down steps, and deduce that these paths are enumerated
by the same Raney numbers.
