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

Shanta Laishram

doi:10.1007/s40993-021-00257-x

paper

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

arXiv:2011.03326 · doi:10.1007/s40993-021-00257-x

Abstract

Let ${U_{n}}_{n \geq 0}$ be a Lucas sequence. Then the equation $∣ U_{n} ∣ = m_{1}! m_{2}! \dots m_{k}!$ with $1 < m_{1} \leq m_{2} \leq \dots \leq m_{k}$ implies $n \in {1, 2, 3, 4, 6, 8, 12}$ . Further the equation $∣ U_{n} ∣ = D_{m_{1}} D_{m_{2}} \dots D_{m_{k}}, D_{m_{i}} \in {B_{m_{i}}, C_{m_{i}}}$ with $1 < m_{1} \leq m_{2} \leq \dots \leq m_{k}$ implies $n \in {1, 2, 3, 4, 6, 8, 12, 16}$ . Here $B_{m}$ is the middle binomial coefficient $(m 2 m)$ and $C_{m}$ is the Catalan number $\frac{1}{m + 1} (m 2 m)$ .

arXiv admin note: text overlap with arXiv:2006.01756