  We are concerned with counting self-conjugate $(s,s+1,s+2)$-core partitions.
A Motzkin path of length $n$ is a path from $(0,0)$ to $(n,0)$ which stays
above the $x$-axis and consists of the up $U=(1,1)$, down $D=(1,-1)$, and flat
$F=(1,0)$ steps. We say that a Motzkin path of length $n$ is symmetric if its
reflection about the line $x=n/2$ is itself. In this paper, we show that the
number of self-conjugate $(s,s+1,s+2)$-cores is equal to the number of
symmetric Motzkin paths of length $s$, and give a closed formula for this
number.
