The main goal of the present paper is to determine the spectral type and solution asymptotics for the Jacobi matrix \(J=J(\{a_n\},\{b_n\})\) with \(a_n=n^\alpha\), \(b_{2n+1}=b(2n+1)^\alpha\), \(b_{2n}=0\), where \(b\) is a positive constant. The authors prove that for \(2/3<\alpha\leq 1\), the spectrum \(\sigma(J)\) on \((-\infty,0)\) is pure absolutely continuous, and for \(0<\alpha\leq 1\), zero is not an eigenvalue of \(J\), and \(\sigma(J)\) on \((0,\infty)\) is pure discrete. Moreover, the eigenvalues \(E_n\) are simple and \(C_1(b) n^\alpha\leq E_n\leq C_2(b) n^\alpha\) with explicit expressions for \(C_k(b)\).