On the properties of Newton - Euler pairs

Abstract. If {an}n=1 and {bn}n=1 are two sequences such that a1 = b1 and bn + a1bn−1 + · · ·+ an−1b1 = nan (n > 1), then we say that (an, bn) is a Newton-Euler pair. In the paper we establish many formulas for Newton-Euler pairs, and then make use of them to obtain new results concerning some special sequences such as p(n), σ(n) and Bn, where p(n) is the number of partitions of n, σ(n) is the sum of divisors of n, and Bn is the n−th Bernoulli number.