Filtering the coefficients of the [2k]-series for Brown-Peterson homology

For non-negative integers k and s let a(k,s) be the s-th coefficient of the universal 2-typical [2(k)]-series. Thus a(k,s) is a polynomial over the 2-local integers on variables nu(1), nu(2),.... We study the structure of this polynomial by using a family of filtrations in the coefficient ring of the Brown-Peterson spectrum. As a result we identify k monomials on a(k,s). The monomials, as well as their 2-divisibility properties, are determined by the dyadic expansion of s + 1.