On the coefficients of an expansion of $(1+1/x)^x$ related to Carleman's   inequality
  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]. 
Yue Hu and Cristinel Mortici
math.CA