Let $Q$ always denote a Dynkin quiver.

Given a connected path algebra $A=kQ$ and a module $M$, is there a useful criterion on $M$ when $End_A(M)$ is again a connected quiver algebra?

Call a module $M$ hereditary in case $End_A(M)$ has this property (which is equivalent that it has global dimension at most 1 and is connected, which are exactly the connected hereditary algebras).

Recall that a module $M$ is called basic in case it has no direct summand of the form $N^2$ for non-zero $N$.

For a given Dynkin quiver $Q$ (with $s(Q)$ denoting the number of points of $Q$) define

$a_{Q,n}:= | \{ M \in mod-kQ | M$ is a hereditary basic module with $n$ indecomposable summands $\} |.$

Of special interest are the numbers

$b_{Q}:= | \{ M \in mod-kQ | M$ is a hereditary basic module with $s(Q)$ indecomposable summands $\} | .$

Can we calculate those numbers in general for Dynkin quivers $Q$?

Special cases are also interesting such as $Q$ being a linear oriented line.

For example we seem to have $a_{Q,2}=\frac{m(m+1)(m+2)(m+7)}{24}$ and $b_Q=2,6,23,114...$ when $Q$ is a linear oriented line with $m+1 \geq 2$ simple modules.

Maybe the question is also interesting when dropping the connecteness condition always.