  We introduce the class of sparse symmetric shifted monomial ideals. These
ideals have linear quotients and their Betti numbers are computed. Using this,
we prove that the symbolic powers of the generalized star configuration ideal
are sequentially Cohen--Macaulay under some mild genericness assumption. With
respect to these symbolic powers, we also consider the Harbourne--Huneke
containment problem and establish the Demailly-like bound.
