Lengyel introduced a sequence of numbers Z_n, defined combinatorially, that satisfy a recurrence where the coefficients are Stirling numbers of the second kind. He proved some 2-adic properties of these numbers. In this paper, we give another recurrence for the sequence Z_n, where the coefficients are Stirling numbers of the first kind. Using this formula, we give another proof of Lengyel's lower bound on the 2-adic valuation of the Z_n. We also resolve some conjectures of Lengyel about the sequence Z_n.

We also define
(a) A new sequence Y_n analogous to Z_n, exchanging the role of Stirling numbers of the first and second kind. We study its 2-adic properties.
(b) Another sequence similar to Lengyel's sequence, and we study its p-adic properties for p ≥ 3.