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This work deals with a class of Jacobi matrices with power-like weights. The main theme is spectral analysis of matrices with zero diagonal and weights $\lambda ₙ:=n^{\alpha }(1+\Delta ₙ)$ where α ∈ (0,1]. Asymptotic formulas for generalized eigenvectors are given and absolute continuity of the matrices considered is proved. The last section is devoted to spectral analysis of Jacobi matrices with qₙ = n + 1 + (-1)ⁿ and $\lambda ₙ=\surd \left(q_{n-1}qₙ\right)$.