Let $A=A_n$ be the representation-finite block of a Schur algebra with $n$ simple modules for $n \geq 2$. Quiver and relations of $A$ can be found in 6.1. of https://arxiv.org/pdf/1607.05965.pdf . Let $A^e=A^{op} \otimes_k A$ be the enveloping algebra. What is the dimension of $\bigoplus_{k=1}^{\infty}{Ext_{A^e}^k(D(A),A)}$ ? Call this number $t_n$. Note that this direct sum is finite since $A^e$ has finite global dimension. For $n=2,3,...,7$ my computer got 4,10,12,16,22,24 which could be https://oeis.org/A095275, which is the sequence (according to OEIS) $b_r=\frac{a_r-3}{4}$ where $a_r$ lists the integers $n$ of the form $4k+3$ for which some of the sums $\sum\limits_{i=1}^{u}{J(\frac{i}{n})}$ are negative where $u$ ranges from 1 to $n-1$ and $J(\frac{i}{n})$ is the Jacobi symbol (interestingly this sequence has also something to do with Motzkin paths, while the stable Auslander-Reiten quiver of this algebra is a infinite periodic Dyck path). This is of course no good evidence but my computer died trying to calculate the $n=8$ case and Im really curious what $t_8$ might be.
Thus my question: What is $t_n$ or at least what is $t_n$ for $n=8,9,10$?
Motivation: This algebra is the simplest class of examples of gendo-symmetric algebras, which are finite dimensional algebras with $D(A) \otimes_A D(A) \cong D(A)$. This is a large class of algebras that include all symmetric algebras, all Schur algebras $S(n,r)$ (for $n \geq r$) and all blocks of category $\mathcal{O}$. I try to find a good ring structure on $\bigoplus_{k=0}^{\infty}{Ext_{A^e}^k(D(A),A)}$ with outer Ext-products using that $D(A) \otimes_A D(A) \cong D(A)$ ($Ext_{A^e}^0(D(A),A)$ is isomorphic to the center of $A$, which is nice). But I can not really calculate any non-trivial examples by hand of this Ext. Another motiation is that this Ext seems to store more information than Hochschild cohomology or homology of $A$ since $D(A)$ has much larger projective dimension compared to $A$ as a bimodule.
It is interesting to note that for $A_n$, $Ext_{A^e}^k(D(A),A)$ is isomorphic to $Ext_{B^e}^k(B,B)$ for $k=0,1,...,4n-6$, which is the Hochschild cohomology of $B=eAe$ (a block of a symmetric group) for e an idempotent such that $eA$ is a minimal faithful projective-injective $A$-module.
