On members of Lucas sequences which are either products of factorials or product of middle binomial coefficients and Catalan numbers

Abstract: Let ${U_n}{n\geq 0}$ be a Lucas sequence. Then the equation $$|U_n|=m_1!m_2!\cdots m_k!$$ with $1<m_1\leq m_2\leq \cdots\leq m_k$ implies $n\in {1,2, 3, 4, 6, 8, 12}$. Further the equation $$|U_n|=D{m_1}D_{m_2}\cdots D_{m_k}, \qquad D_{m_i}\in {B_{m_i}, C_{m_i}}$$ with $1<m_1\leq m_2\leq \cdots\leq m_k$ implies $n\in {1,2, 3, 4, 6, 8, 12, 16}$. Here $B_m$ is the middle binomial coefficient $\binom{2m}{m}$ and $C_m$ is the Catalan number $\frac{1}{m+1}\binom{2m}{m}$.