The coefficient of $x^j$ in $(T_n(x)\bmod (x^r-1))$ equals the coefficient of $t^{n+r-j-1}$ in $$\frac{(1+t^2)^{r-j}}{2^{r-j}} \frac{((1+t^2)^{r-1}t - 2^{r-1}t^{r-1})}{((1+t^2)^r - 2^rt^r)}.$$ This coefficient can be explicitly computed as $$\sum_{k\geq 0} 2^{rk-r+j} \left( \binom{r-1-j-rk}{\frac{n+r-j-2-rk}{2}} - 2^{r-1}\binom{-j-rk}{\frac{n-j-rk}{2}}\right).$$ (here the binomial coefficients are zero whenever their lower indices are noninteger or negative)
