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In this paper, we give some explicit expressions of $\\sum_{l=0}^{2 r-3}(-1)^l\\binom{2 r-3}{l}\\sum_{j_1+\\cdots+j_r=n-2 l\\atop j_1,\\dots,j_r\\ge 1}B_{j_1}\\cdots B_{j_r}$ and $\\sum_{j_1+\\cdots+j_r=n\\atop j_1,\\dots,j_r\\ge 1}B_{j_1}\\cdots B_{j_r}$. 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