I want to calculate integral of polynomial $x \mapsto x^n$ over set $C_m$ i.e.
$$I = \int_{C_m} x^n dx.$$
Set $C_m$ is defined recursively with $C_0 = [0, 1]$ and for $m > 0$
$$C_m = \frac{C_{m - 1}}{3}\cup\frac{C_{m - 1}+2}{3}.$$
As a result, I get
$$I = \frac{1}{n + 1}\sum_{\alpha \in \{0, 1\}^m} \left( \left(\frac{1}{3^m} +\sum_{k = 1}^{m} \frac{2 \alpha_k}{3^k}\right)^{n+1} - \left(\sum_{k = 1}^{m} \frac{2 \alpha_k}{3^k}\right)^{n+1}\right). $$
Is there a way to simplify $I$?