Undescribed exponential growth rates

There is a function $ d_{(A,M )} (n) $ called growth rate that is defined for an arbitrary finite set $A$ with a set of operations $M$ defined on it. It characterizes the strength of given operations. It has been proved that growth rate is either $ O(n^k) $ for some $ k \in \mathbb{N} $, either $ 2^{\Theta(n)} $. We research classes of exponential growth rates that appear after splitting the class with asymptotic bound in the exponent to classes with outward asymptotic bounds. We show that there exists a pair $(A, M)$ with the growth rate $ d_{(A,M)}(n) \in \Theta (n^k \cdot c^n)$ for arbitrary predefined natural numbers $k$ and $c$. In addition, if $c > k + 1$ then there exists a pair $(A, M)$ with the growth rate $d_{(A,M)}(n) \in \Theta (\log n \cdot n^k \cdot c^n)$.