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 $\frac{ 1}{ m+1}\binom{ 2m}{ m}$.
Categories: math.NT