In this paper, we consider a multi-cell multi-band downlink where the base station (BS) in each cell has multiple transmit antennas. Each cell has one active mobile station (MS) with a single receive antenna and treats interference from the other cells as noise. There is a sum transmit power constraint for each BS over all the bands. An alternating maximization (AM) algorithm is proposed to determine the optimal power allocation among the bands and the optimal beamforming vectors for each BS in each band. This algorithm can be implemented in a distributed manner with limited exchange of interference constraints between the BSs, and only local channel state information at each BS. The proposed algorithm alternates between: (1) weighted sum-rate (WSR) optimization for the beamformers in each band for a given power allocation, and (2) optimal power allocation across bands for a given set of beamformers. For the 2-cell and 3-cell settings the WSR optimization in each band is significantly simplified using analytical solutions for the sub-problems. The power allocation across bands for a given set of beamformers is obtained analytically in all cases. Numerical results show good convergence properties and significant performance gain using the proposed AM algorithm compared to: (i) equal power allocation across bands and weighted sum-rate optimization in each band, (ii) zero-forcing (ZF) beamforming, and (iii) maximal ratio transmission (MRT) beamforming.