The number of triangulations of the n-cube is a distant cousin of Catalan numbers as they both count triangulations of some set of points. - Lionel Pournin, Mar 06 2014

REFERENCES

Lionel Pournin, The flip-graph of the 4-dimensional cube is connected, Discrete Comput. Geom. 49 (2013), no. 3, 511--530.MR3038527