  In this note we study the generalized Hilbert series operator $H_{\mu}$,
induced by a positive Bore measure $\mu$ on $[0, 1)$, between weighted sequence
spaces. We characterize the measures $\mu$ for which $H_{\mu}$ is bounded
between different sequence spaces. Finally, for certain special measures, we
obtain the sharp norm estimates of the operators and establish some new
generalized Hilbert series inequalities with the best constant factors.
