  For any integer $k\geq 1,$ define $L_k: \mathbb{R}^\mathbb{N}\to
\mathbb{R}^\mathbb{N}$ by $(a_n)_{n\in\mathbb{N}}\mapsto
(a'_n)_{n\in\mathbb{N}}$ where $a'_n=\det(a_{n+i+j})_{i,j=0}^{k-1}$.
Previously, Zhu showed that $L_k$ preserves the Stieltjes moment (SM) property
of sequences (Proc. Am. Math. Soc., 2019). The proof used the characterization
of SM sequences in terms of positive semidefinite Hankel matrices. In this
note, we give another proof by viewing SM sequences as weighted enumerations of
closed walks on $\mathbb{N}$. Our proof is essentially a double-counting
argument that views a $k$-tuple of non-crossing Dyck paths as a single closed
walk on some bipartite subgraph of $\mathbb{N}^k.$
