We denote $\log_2$ the logarithm function in base 2. For $n \geq 1$, we set $S_k(n) = \sum_{m=2}^{n} D_k(m)$ where $D_k(m)$ is the number of ways to decompose the integer $m$ into a product of $k$ factors greater than or equal to 2, the order of these factors being important. Deduce from the previous question that
$$\chi_n(\lambda) = (\lambda - 1)^n - \sum_{k=1}^{\left\lfloor \log_2 n \right\rfloor} (\lambda - 1)^{n-k-1} S_k(n).$$