Journal of Integer Sequences, Vol. 17 (2014), Article 14.7.3

p-adic Properties of Lengyel's Numbers

D. Barsky
7 rue La Condamine
75017 Paris

J.-P. Bézivin
1, Allée Edouard Quincey
94200, Ivry-sur-Seine


Lengyel introduced a sequence of numbers Zn, 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 Zn, 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 Zn. We also resolve some conjectures of Lengyel about the sequence Zn.

We also define
(a) A new sequence Yn analogous to Zn, 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.

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(Concerned with sequence A005121.)

Received January 24 2014; revised version received June 2 2014; June 16 2014. Published in Journal of Integer Sequences, June 17 2014.

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