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

TL;DR

This paper classifies Lucas sequence members that are either products of factorials or products of middle binomial coefficients and Catalan numbers, identifying specific n-values where these equalities hold.

Contribution

It provides a complete characterization of Lucas sequence members expressible as such products, extending previous results on special number forms.

Findings

n belongs to {1,2,3,4,6,8,12} for factorial products

n belongs to {1,2,3,4,6,8,12,16} for binomial and Catalan products

No other Lucas sequence members fit these product forms outside these n-values

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 $\frac{1}{m+1}\binom{2m}{m}$.