Suppose you are given a power series $P=\sum\_{i=0}^\infty{a\_nt^n}$. I am primarily concerned with those power series coming from rational functions of the form 
$$ \frac{1}{\prod\_{i=1}^k{(1-t^{\alpha\_i})}}.$$ My motivation comes from looking at Hilbert series of noetherian freely generated subrings of polynomial rings. In general, I know $P$ and its expression as a rational function. I then consider the power series $P'=\sum\_{i=0}^\infty{a\_n^\kappa t^n}$ for some $\kappa\in\mathbb{Z}\_{>0}$. I am interested in determining facts about the rational function that describes this power series (there should be one as it is a Hilbert series of a ring). Of greatest interest to me is the degree of the polynomial appearing in the denominator.



Raising coefficients of a power series to some power
It's a general fact about any rational power series


$$\sum\_{n=0}^\infty a\_n t^n = {p(t) \over \prod\_{i=1}^r (1-\gamma\_i t)^{d\_i}}$$


(where $p(t)$ is some polynomial of degree less than $\sum\_{i=1}^r d\_i$ and the $\gamma\_i$ are distinct) that


$$a\_n = \sum\_{i=1}^r p\_i(n)\gamma\_i^n$$


for some polynomials $p\_i$ of degree less than $d\_i$. Conversely, any $a\_n$ that can be expressed this way are the coefficients of a rational power series. See for example Theorem 4.1.1 in Richard Stanley's book *Enumerative Combinatorics*. From this it is immediate that $a\_n^\kappa$ are the coefficients of a rational power series, as you say, and moreover by chasing through the proof of the theorem you can construct the denominator that you are interested in. I think that this shows that the degree will go up by at most a factor of $\kappa$, but to compute the degree exactly, you may have to keep careful track of which roots of unity occur with what multiplicities.


