Sequences Derived from The Symmetric Powers of $\{1,2,\ldots,k\}$

For a fixed integer $k$, we define a sequence $A_k=(a_k(n))_{n\geq0}$ and a corresponding sparse subsequence $S_k$ using the cardinality of the $n$-th symmetric power of the set $\{1,2,\ldots, k\}$. For $k\in\{2,\dots,8\}$, we find recursive formulas for $S_k$, and show that the values $a_{k}(0)$, $a_{k}(1)$, and $a_{k}(3)$ are sufficient for constructing $A_{k}$.