Sums of infinite series involving the Riemann zeta function II

We determine for all natural numbers p the exact value of the converging series ∑n=1∞np(ζ(2n)-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sum _{n=1}^\infty n^p (\zeta (2n)-1)}$$\end{document}. Two recursive formulas are given, too. The cases p=1,2,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=1,2,3$$\end{document} are done right at the beginning to illustrate the method used to derive these formulas.