Let $I$ be a finite set. For each $i \in I$ let $(x^{(i)}_n)_{n \in \mathbb{N}}$ be monotonically increasing sequence of strictly positive integers. Under what conditions on the sequences $(x^{(i)}_n)_{n \in \mathbb{N}}$ do there exist weights $p_i\ge 0$ (that is $\sum_{i \in I}p_i = 1$) so that we have the equality
$$ \Big(\sum_{i \in I} p_i x^{(i)}_1 \Big)^n = \sum_{i \in I} p_ix_n^{(i)} $$ for all $n \in \mathbb{N}$.
I know, for example, that we get equality if $p_i = 1/|I|$ and for each $i$ and $n$ we have $x_n^{(i)} = a^n$ for some positive integer $a$.
In general, is there a characterisation of the allowable sequences?