Differential production cross sections of prompt J/psi and psi(2S) charmonium and Upsilon(nS) (n = 1, 2, 3) bottomonium states are measured in proton-proton collisions at root s = 13 TeV, with data collected by the CMS detector at the LHC, corresponding to an integrated luminosity of 2.3 fb(-1) for the J/psi and 2.7 fb(-1) for the other mesons. The five quarkonium states are reconstructed in the dimuon decay channel, for dimuon rapidity vertical bar y vertical bar < 1.2. The double-differential cross sections for each state are measured as a function of y and transverse momentum, and compared to theoretical expectations. In addition, ratios are presented of cross sections for prompt psi(2S) to J/psi, Upsilon(2S) to Upsilon(1S), and Upsilon(3S) to Upsilon(1S) production. (C) 2018 The Author(s). Published by Elsevier B.V.