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

Let $$\{U_n\}_{n\ge 0}$$ be a Lucas sequence. We show that if $$U_n$$ is a product of factorials, then $$n\in \{1,2, 3, 4, 6, 8, 12\}$$ . Further we show that if $$U_n$$ is a product of Catalan numbers or the middle binomial coefficients, then $$n\in \{1,2, 3, 4, 6, 8, 12, 16\}$$ . Here the $$m-$$ th middle binomial coefficient is $$B_m=\left( {\begin{array}{c}2m\\ m\end{array}}\right) $$ and the $$m-$$ th Catalan number is $$C_m=\frac{1}{m+1}\left( {\begin{array}{c}2m\\ m\end{array}}\right) $$ .