as a distinguished segment. We may use a star $*$ to mark the distinguished segment. We call a composition with a distinguished segment a [*rooted*]{} composition of a Dyck path. Then $ja_{i,j}$ equals the number of rooted compositions of Dyck paths of length $2i$ that contain $j$ segments.

There is a bijection $\phi$ between the set of rooted compositions of Dyck paths of length $2i$ and the set of sequences of length $i-1$ on four letters.

Let $(P_1, P_2, \ldots, P_j)$ be a rooted composition of a Dyck path $P$ of length $2i$. We proceed to construct a sequence of length $i-1$ on the elements $\{ 1, 2,3, 4\}$. We now recursively define a map $\phi$ from rooted compositions of a Dyck path $P$ of length $2i$ to sequences of length $i-1$ on $\{
1, 2, 3, 4\}$. For $i=1$, $P$ is unique, and the sequence is set to be the empty sequence. We now assume that $i>1$. Let $(P_1,
\ldots, P_t^*, \ldots, P_j)$ be a rooted composition of $P$ with $P_t^*$ being the distinguished segment.

We have the following cases.

1.  $P_1=UD$ and $t=1$. Then we set $\phi(P)=1\, \phi(P_2^*, P_3, \ldots, P_j)$.

2.  $P_1=UD$ and $t\not=1$. Then we set 