We discuss how to calculate cross sections as well as rapidity, transverse momentum and energy distributions of $\nu_{\tau}$ and ${\overline \nu}_{\tau}$ produced from the direct $D_s^{\pm} \to \nu_{\tau}/{\overline \nu}_{\tau}$ and chain $D_s^{\pm} \to \tau^+/\tau^- \to \nu_{\tau}/{\overline \nu}_{\tau}$ decays in $p\!+^{96}\!\mathrm{Mo}$ scattering with proton beam $E_{\mathrm{lab}}$ = 400 GeV \textit{i.e.} at $\sqrt{s}_{NN}$ = 27.4 GeV. The $\tau$ decays are simulated with the help of the \textsc{Tauola} code and include multiple decay channels of $\tau$ in amounts proportional to their branching ratios... In our calculations we include $D_s^{\pm}$ from charm fragmentation $c \to D_s^{+}$ and $\bar c \to D_s^-$ as well as those from subleading fragmentation of strange quarks/antiquarks $s \to D_s^-$ and $\bar s \to D_s^+$. The $s \ne \bar s$ asymmetry of the strange quark content of proton is included. The different contributions to $D_s^{\pm}$ and $\nu_{\tau} / {\overline \nu}_{\tau}$ are shown explicitly. We discuss and quantify a not discussed so far effect of asymmetries for production of $\nu_{\tau}$ and ${\overline \nu}_{\tau}$ caused by subleading fragmentation mechanism and discuss related uncertainties. A potential measurement of the asymmetry is discussed. Estimates of a number of observed $\nu_{\tau} / \overline{\nu}_{\tau}$ in the $\nu_{\tau} / \overline{\nu}_{\tau} +^{208}\!\mathrm{Pb}$ reaction, with 2m long target are given with the help of the NuWro program. We refer also to the production of the high-energy (anti)neutrinos in the atmosphere. (read more)