Sequences Derived from the Symmetric Powers of {1, 2, …, k}

For a fixed integer k, we define a sequence A_k = (a_k (n))_n≥0 and a corresponding sparse subsequence S_k using the cardinality of the n-th symmetric power of the set {1, 2, …, k}. For k ∈ {2, …, 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.