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

The positive part $U_q^+$ of $U_q(\widehat{\mathfrak{sl}}_2)$ 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 $D(t)=\sum_{n \in \mathbb{N}}D_nt^n$ is the multiplicative inverse of the generating function $\tilde{G}(t)=\sum_{n \in \mathbb{N}}\tilde{G}_nt^n$ 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.