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 $s\ge 1$, we prove that the sequence $(\operatorname{odd}(t_m(n))\bmod 2^s)_{n\ge 0}$ is $2$-automatic when $m=2^r$, $r\ge 1$, or $m=3\cdot 2^r$, $r\ge 2$. In both families, $ν_2(t_m(n))=ν_2\binom{n+m-1}{m-1}$. The valuation identity for $m=2^r$ is known; our proof recovers it and also establishes the automaticity assertion. For $m=6$ we prove $ν_2(t_6(n))=ν_2\binom{n+5}{5}+\mathbf{1}_{n\equiv 3\pmod 4}$ and show that its odd parts modulo every $2^s$ are $2$-automatic.