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Let $F(x)=\prod_{n=0}^{\infty}(1-x^{2^{n}})$ be the generating function forthe Prouhet-Thue-Morse sequence $((-1)^{s_{2}(n)})_{n\in\N}$. In this paper weinitiate the study of the arithmetic properties of coefficients of the powerseries expansions of the function $$F_{t}(x)=F(x)^{t}=\sum_{n=0}^{\infty}f_{n}(t)x^{n}. $$ For $t\in\N_{+}$ thesequence $(f_{n}(t))_{n\in\N}$ is the Cauchy convolution of $t$ copies of theProuhet-Thue-Morse sequence. For $t\in\Z_{<0}$ the $n$-th term of the sequence$(f_{n}(t))_{n\in\N}$ counts the number of representations of the number $n$ asa sum of powers of 2 where each summand can have one among $-t$ colors.  Among other things, we present a characterization of the solutions of theequations $f_{n}(2^k)=0$, where $k\in\N$, and $f_{n}(3)=0$. Next, we presentthe exact value of the 2-adic valuation of the number $f_{n}(1-2^{m})$ - aresult which generalizes the well known expression concerning the 2-adicvaluation of the values of the binary partition function introduced by Eulerand studied by Churchhouse and others.

Arithmetic properties of coefficients of power series expansion of  $\prod_{n=0}^{\infty}\left(1-x^{2^{n}}\right)^{t}$ (with an Appendix by  Andrzej Schinzel)