  The goal of this short note is to study the secant varieties of the triple
Segre product of type (1,a,b) by means of the standard tools of combinatorial
commutative algebra. We reprove and extend to arbitrary characteristic results
of Landsberg and Weyman regarding the defining ideal and the Cohen-Macaulay
property of the secant varieties. Furthermore for these varieties we compute
the degree and give a bound for their Castelnuovo-Mumford regularity which is
sharp in many cases.
