Let $(a_n)$ be a sequence of non-negative real numbers and assume that the resulting power series defines a rational function

$$\sum_{n=0}^\infty a_n x^n = \dfrac{f(x)}{(1-x^{k_1})\cdots (1-x^{k_d})}$$ where $k_1,...,k_d>0$ are integers and $f(x)$ is a real polynomial s.t. $f(1) \neq 0$. It is not hard to show that $$\frac{f(1)}{k_1 \cdots k_d}\le \limsup \frac{a_n}{n^{d-1}} \cdot (d-1)! \le f(1)$$

As a special case we obtain for example $\limsup = f(1)$ if $k_1=\cdots k_d = 1$ (this is in fact not only the limsup but even the limit of the sequence).

**Questions:** 1) Are there known formulas or better estimates for the $\limsup$ above in terms of $f$ and $k_1,...,k_d$ ?

2) Are there particular techniques, that can be used to obtain good estimates (the one above is simply based on the binomial series for $(1-x)^{-d}$).

**Background:** Such rational functions occur as Poincaré series of graded Noetherian algebras where $a_n$ is the dimension of the subspace of homogeneous lements of degree $n$. I'm trying to relate this quantity to the rational function.