2-adic Valuations of Coefficients of the Fifth and Ninth Powers of the Thue--Morse Generating Function

Let $T(x)={\prod }_{k=0}^{\mathrm{\infty }}(1-x^{2^k})$ be the generating function of the Thue–Morse sequence, and write $T(x{)}^m={\sum }_{n\ge 0}{t}_m(n)x^n$. We prove exact formulas for the $2$-adic valuations of the coefficients $t_5(n)$ and $t_9(n)$: $${\nu }_2(t_5(4n+j))=4\lceil \frac{{\nu }_2(n+1)}{2}\rceil -({\nu }_2(n+1)2),j\in \{0,1,2,3\},$$ $${\nu }_2(t_9(8n+j))=5\lceil \frac{{\nu }_2(n+1)}{2}\rceil -2({\nu }_2(n+1)2),j\in \{0,1,\dots ,7\}\mathrm{.}$$ These formulas confirm Conjecture~5.2 of Gawron–Miska–Ulas~\cite{ga} for $m=5$ and $m=9$, and imply that $t_5(n)\mathrm{\ne }0$ and $t_9(n)\mathrm{\ne }0$ for every $n\ge 0$. A key structural ingredient is a closed-form formula for the determinant of a family of matrices with binomial-coefficient entries.