Are $n \times n$ matrices of the form
$$\pmatrix{1&1&1&1 \cr x&1&1&1 \cr x&x&1&1 \cr x&x&x&1}$$
studied anywhere? I am interested in the structure of the matrix obtained by multiplying a bunch of these together.

These are special Töplitz matrices. Maybe you should look at their inverses: it has -1/(x-1) on the main diagonal, 1/(x-1) on the superdiagonal and x/(x-1) in the lower left corner. All the other entries are zero.
– Martin RubeyApr 18 '13 at 16:05

They also happen to be semiseparable.
– Federico PoloniApr 18 '13 at 18:16

Thank you Martin and Federico. $1/(1-x)$ resonates well with my problem!
– Rodrigo A. PérezApr 18 '13 at 22:11