  Let $K_n$ denote the number of types of a sample of size $n$ taken from an
exchangeable coalescent process ($\Xi$-coalescent) with mutation. A
distributional recursion for the sequence $(K_n)_{n\in{\mathbb N}}$ is derived.
If the coalescent does not have proper frequencies, i.e., if the characterizing
measure $\Xi$ on the infinite simplex $\Delta$ does not have mass at zero and
satisfies $\int_\Delta |x|\Xi(dx)/(x,x)<\infty$, where $|x|:=\sum_{i=1}^\infty
x_i$ and $(x,x):=\sum_{i=1}^\infty x_i^2$ for $x=(x_1,x_2,...)\in\Delta$, then
$K_n/n$ converges weakly as $n\to\infty$ to a limiting variable $K$ which is
characterized by an exponential integral of the subordinator associated with
the coalescent process. For so-called simple measures $\Xi$ satisfying
$\int_\Delta\Xi(dx)/(x,x)<\infty$ we characterize the distribution of $K$ via a
fixed-point equation.
