Stern polynomials

Stern polynomials Bk(t), k⩾0, t∈R, are introduced in the following way: B0(t)=0, B1(t)=1, B2n(t)=tBn(t), and B2n+1(t)=Bn+1(t)+Bn(t). It is shown that Bn(t) has a simple explicit representation in terms of the hyperbinary representations of n−1 and that equals the number of 1's in the standard Gray code for n−1. It is also proved that the degree of Bn(t) equals the difference between the length and the weight of the non-adjacent form of n.