On monotonicity of some combinatorial sequences (1208.3903v8)

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 $(F_n){n\ge0}$ 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 $H_n^{(m)}=\sum{k=1}^n 1/k^m\ (n=1,2,3,\ldots)$, we show that $(\root{n+1}\of{H^{(m)}{n+1}}/\root{n}\of{H^{(m)}_n}){n\ge3}$ is strictly increasing.