Powers of the Thue--Morse Series: 2-Adic Valuations and Automatic Odd Parts

Let $T(x)=\prod_{j\ge 0}(1-x^{2^j})=\sum_{n\ge 0}t(n)x^n$ be the Thue--Morse generating function, and write $T(x)^m=\sum_{n\ge 0}t_m(n)x^n$ for a positive integer $m$. For a nonzero integer $a$, let $ν_2(a)$ be the exponent of $2$ in $a$ and put $\operatorname{odd}(a)=a/2^{ν_2(a)}$. For every $s\ge