On arithmetic properties of binary partition polynomials.

Let the polynomial b n (t) be defined as the n -th coefficient in the power series expansion (in variable x) of the function B (t , x) = ∏ n = 0 ∞ 1 1 − t x 2 n = ∑ n = 0 ∞ b n (t) x n. The polynomial b n (t) = ∑ i = 0 n a (i , n) t i has the following combinatorial interpretation: the i -th coefficient a (i , n) counts the number of representations of n as sums of exactly i powers of 2. The b n (t) can be seen as a polynomial analogue of the number b n (1) , which counts the number of binary partitions of n. In the present paper we obtain several results concerning arithmetic properties of the polynomials b n (t) as well as its coefficients. Moreover, we show an interesting connection between coefficients of b n (t) and the number counting so called s -partitions, i.e., the representations of n as sums of numbers of the form 2 k − 1. [ABSTRACT FROM AUTHOR]