A Note on the Generalized Bernoulli Sequenses

Abstract

Let $n,s$ be positive integers, and let $r=1+\frac{1}{s}$. In this note we prove that if the sequence $\{a_n(r){\}}_{n=1}^{\mathrm{\infty }}$ satisfies $r{a}_n(r)=\sum _{k=1}^n(\genfrac{}{}{0ex}{}{n}{k})a_k(r),n>1$, then $a_n(r)=n{a}_1(r)[(n-1)!/(s+1)(logr{)}^n+1/r(s+1)]$. Moreover, we obtain a related combinatorial identity.