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

Zhao Shen, Xinping Wang

Let $T(x)={\prod }_{k=0}^{\infty }(1-x^{2^k})$ 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 $2$-a…