  In this paper, the production rates of quarkonia eta_c, J/psi, eta_b,
Upsilon, B_c and B_c^* through W boson decay at the LHC are calculated, at the
leading order in both the QCD coupling constant and in v, the typical velocity
of the heavy quark inside of mesons. It shows that a sizable number of
quarkonia from W boson decay will be produced at the LHC. Comparison with the
predictions by using quark fragmentation mechanism is also discussed. Results
show that, for the charmonium production through W decay, the difference
between predictions by the fragmentation mechanism and complete leading order
calculation is around 3%, and it is insensitive to the uncertainties of
theoretical parameters; however, for the bottomonium and B_c^(*) productions,
the difference cannot be ignored as the fragmentation mechanism is less
applicable here due to the relatively large ratio mb/mw.
