  If the $X(3872)$ is a weakly bound charm-meson molecule, it can be produced
by the creation of $D^{*0} \bar{D}^0$ or $D^{0} \bar{D}^{*0}$ at short
distances followed by the formation of the bound state from the charm-meson
pair. The $X$ can also be produced by the creation of $D^* \bar{D}^*$ at short
distances followed by the rescattering of the charm-meson pair into $X \pi$. At
a high-energy hadron collider, the prompt cross section from this mechanism has
a narrow peak in the $X \pi$ invariant mass distribution near the $D^* \bar
D^*$ threshold from a charm-meson triangle singularity. An order-of-magnitude
estimate of the ratio of the cross section for producing the peak in the $X
\pi^\pm$ distribution to the cross section for producing $X$ without an
accompanying pion suggests that the peak may be observable at the LHC.
