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

Abstract

Let $\{U_n\}_{n\ge 0}$ 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 $C_m=\frac{1}{m+1}\left( {\begin{array}{c}2m\\ m\end{array}}\right)$.