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

For a fixed integer $k$, we define a sequence $A_k=(a_k(n){)}_{n\ge 0}$ and a corresponding sparse subsequence $S_k$ using the cardinality of the $n$-th symmetric power of the set $\{1,2,\dots ,k\}$. 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 $A_k$.