  We present a new calculation of the energy distribution of high-energy
neutrinos from the decay of charm and bottom hadrons produced at the Large
Hadron Collider (LHC). In the kinematical region of very forward rapidities,
heavy-flavor production and decay is a source of tau neutrinos that leads to
thousands of { charged-current} tau neutrino events in a 1 m long, 1 m radius
lead neutrino detector at a distance of 480 m from the interaction region. In
our computation, next-to-leading order QCD radiative corrections are accounted
for in the production cross-sections. Non-perturbative intrinsic-$k_T$ effects
are approximated by a simple phenomenological model introducing a Gaussian
$k_T$-smearing of the parton distribution functions, which might also mimic
perturbative effects due to multiple initial-state soft-gluon emissions. The
transition from partonic to hadronic states is described by phenomenological
fragmentation functions. To study the effect of various input parameters,
theoretical predictions for $D_s^\pm$ production are compared with LHCb data on
double-differential cross-sections in transverse momentum and rapidity. The
uncertainties related to the choice of the input parameter values, ultimately
affecting the predictions of the tau neutrino event distributions, are
discussed. We consider a 3+1 neutrino mixing scenario to illustrate the
potential for a neutrino experiment to constrain the 3+1 parameter space using
tau neutrinos and antineutrinos. We find large theoretical uncertainties in the
predictions of the neutrino fluxes in the far-forward region. Untangling the
effects of tau neutrino oscillations into sterile neutrinos and distinguishing
a 3+1 scenario from the standard scenario with three active neutrino flavours,
will be challenging due to the large theoretical uncertainties from QCD.
