On monotonicity of some combinatorial sequences

Abstract

We confirm Sun's conjecture that $(\root{n+1}\of{F_{n+1}}/\root{n}\of{F_n})_{n\ge 4}$ is strictly decreasing to the limit 1, where $(F_n{)}_{n\ge 0}$ is the Fibonacci sequence. We also prove that the sequence $(\root{n+1}\of{D_{n+1}}/\root{n}\of{D_n})_{n\ge3}$ is strictly decreasing with limit $1$, where $D_n$ is the $n$-th derangement number. For $m$-th order harmonic numbers $H_n^{(m)}={\sum }_{k=1}^{n}1/k^m(n=1,2,3,\dots )$, we show that $(\root{n+1}\of{H^{(m)}_{n+1}}/\root{n}\of{H^{(m)}_n})_{n\ge3}$ is strictly increasing.

10 pages

References in corpus (1)

• Conjectures involving arithmetical sequences

Cited by in corpus (3)

• Conjectures involving arithmetical sequences
• Higher order log-monotonicity of combinatorial sequences
• On some monotonic combinatorial sequences conjectures