In the paper, the main purpose is to define and to investigate the sequence space $Ces(p,s)$ and to determine the matrices of classes $(Ces(p,s),\ell_\infty)$ and $(Ces(p,s),c)$ where $\ell_\infty\infty$ and $c$ are respectively the space of bounded and convergent complex sequences and for $p=(p_r)$ with $\inf p_r>0$, the space $Ces(p,s)$ is defined for $s>0$ by \[ Ces(p,s)=\bigg\{x=(x_k):um^ıfty_{r=0}(2^r)^{-s}\Big(\frac1{2^r}um_r|x_k|\Big)^{p_r}<ıfty\bigg\} \] where $\sum_r$ denotes a sum over the ranges $2^r\leq k<2^{r+1}$. These spaces i.e. $Ces(p,s)$ can be viewed as $Ces(p)$ spaces with weights, generalizing $Ces(p)$ spaces.