  We consider standard $\La$-coalescents (or coalescents with multiple
collisions) with a non-trivial "Kingman part". Equivalently, the driving
measure $\Lambda$ has an atom at $0$; $\Lambda(\{0\})=c>0$. It is known that
all such coalescents come down from infinity. Moreover, the number of blocks
$N_t$ is asymptotic to $v(t) = 2/(ct)$ as $t\to 0$. In the present paper we
investigate the second-order asymptotics of $N_t$ in the functional sense at
small times. This complements our earlier results on the fluctuations of the
number of blocks for a class of regular $\La$-coalescents without the Kingman
part. In the present setting it turns out that the Kingman part dominates, and
the limit process is a Gaussian diffusion, as opposed to the stable limit in
our previous work.
