Arithmetic properties of the sequence of degrees of Stern polynomials and related results

Let $B_{n}(t)$ be a $n$-th Stern polynomial and let $e(n)=\op{deg}B_{n}(t)$ be its degree. In this note we continue our study started in \cite{Ul} of the arithmetic properties of the sequence of Stern polynomials and the sequence $\{e(n)\}_{n=1}^{\infty}$. We also study the sequence $d(n)=\op{ord}_{t=0}B_{n}(t)$. Among other things we prove that $d(n)=\nu(n)$, where $\nu(n)$ is the maximal power of 2 which dividies the number $n$. We also count the number of the solutions of the equations $e(m)=i$ and $e(m)-d(m)=i$ in the interval $[1,2^{n}]$. We also obtain an interesting closed expression for a certain sum involving Stern polynomials.