Count the number of ways elements in the interval [1,m] can be partitioned into k sum free sets

The Schur number $S(k)$ is the largest integer $n$ for which the interval $[1,n]$ can be partitioned into $k$ sum-free sets. 1

Now consider a related question, the number of ways elements in the interval $[1,m]$ can be partitioned into $k$ sum free sets, call it $S(k, m)$ for $m \le S(k)$ (order doesn't matter, so there's only one way to place $1$, only one way to place $1,2$, etc depending on the number of partitions after that). Of course, there is a maximal $m$ such that $S(k,m)$ will yield $S(k)$ and above which should return $0$. (This is different from http://oeis.org/A007865).

I'm looking for more information for this but can't seem to find anything on OEIS. Some of the first few values are:

S(3, 1): 1
S(3, 2): 1
S(3, 3): 3
S(3, 4): 5
S(3, 5): 11
S(3, 6): 20
S(3, 7): 43
S(3, 8): 48
S(3, 9): 91
S(3, 10): 50

S(4, 1): 1
S(4, 2): 1
S(4, 3): 3
S(4, 4): 6
S(4, 5): 19
S(4, 6): 54
S(4, 7): 181
S(4, 8): 426
S(4, 9): 1395
S(4, 10): 2964


But perhaps there's an error with my algorithm and my counts are wrong, which is why I can't find anything.