  In this note, we present new properties for a sequence arising in some
refinements of Carleman's inequality. Our results extend some results of Yang
[Approximations for constant e and their applications J. Math. Anal. Appl. 262
(2001) 651-659] and Alzer and Berg [some classes of completely monotonic
functions Ann. Acad. Sci. Fennicae 27(2002) 445-460].
