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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.

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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 Rubey Apr 18 '13 at 16:05
    
They also happen to be semiseparable. –  Federico Poloni Apr 18 '13 at 18:16
    
Thank you Martin and Federico. $1/(1-x)$ resonates well with my problem! –  Rodrigo A. Pérez Apr 18 '13 at 22:11
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