A generating function associated with the alternating elements in the positive part of $U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_2)$

The positive part $U_q^{+}$ of $U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_2)$ admits an embedding into a $q$-shuffle algebra. This embedding was introduced by M. Rosso in 1995. In 2019, Terwilliger introduced the alternating elements $\{W_{-n}{\}}_{n\in \mathbb{N}}$, $\{W_{n+1}{\}}_{n\in \mathbb{N}}$, $\{G_{n+1}{\}}_{n\in \mathbb{N}}$, $\{{\tilde{G}}_{n+1}{\}}_{n\in \mathbb{N}}$ in $U_q^{+}$ using the Rosso embedding. He showed that the alternating elements $\{W_{-n}{\}}_{n\in \mathbb{N}}$, $\{W_{n+1}{\}}_{n\in \mathbb{N}}$, $\{{\tilde{G}}_{n+1}{\}}_{n\in \mathbb{N}}$ form a PBW basis for $U_q^{+}$, and he expressed $\{G_{n+1}{\}}_{n\in \mathbb{N}}$ in this alternating PBW basis. In his calculation, Terwilliger used some elements $\{D_n{\}}_{n\in \mathbb{N}}$ with the following property: the generating function $D(t)=\sum_{n \in $\mathbb{N}$}D_nt^n$isthemultiplicativeinverseofthegeneratingfunction$\tilde{G}(t)=\sum_{n \in $\mathbb{N}$}\tilde{G}_nt^n$where$\tilde{G}_0=1$. Terwilliger defined $\{D_n{\}}_{n\in \mathbb{N}}$ recursively; in this paper, we will express $\{D_n{\}}_{n\in \mathbb{N}}$ in closed form.