This question is a modification of Number of tilting modules and was suggested by a comment of Dag Madsen. I wondered about this question too some time ago, but did not do much with it since my computer is too slow to test things for larger numbers.

Let $A=A_n$ be the algebra of upper triangular matrices over a field $K$ with $n$ simple modules. Let $J$ be the Jacobson radical of $A_n$ and $B_{n,l}:=A_n /J^l$ for some $n-1 \geq l \geq 2$. A generalised tilting module $M$ is a module with $n$ direct summands such that $Ext^i(M,M)=0$ for all $i=1,2,...,g$ ,where $g$ is the global dimension of the algebra. You can find a formula for the global dimension of those algebras in https://arxiv.org/pdf/1707.03996.pdf proposition 6.8.

How many generalised tilting modules are there in $B_{n,l}$? Let $b_{n,l}$ denote this sequence.

Some examples:

-$b_{n,n}$ are the Catalan numbers.

-$b_{n,2}=n$

-$b_{n,3}$ for $ n \geq 3$ might be 5, 9, 15, 22, 30, 40, 51, 63, see https://oeis.org/A022941 .

-$b_{n,4}$ starts with 14,28,52,91 for $n \geq 4$ and already does not exist in the OEIS.

-$b_{n,n-1}$ might be more interesting and starts with 3,9,28,90,297 for $n \geq 3$ and might be the Number of standard tableaux of shape (n+1,n-1), see https://oeis.org/A071724

-$b_{n,n-2}$ starts with 4,15,52,179 for $n \geq 4$ and again seems to be not in the OEIS.