Factorization of Lower Triangular Toeplitz Matrix

Is it possible to factorize a lower triangular Toeplitz Matrix, using binary matrices and a vector of elements?

For example, the Toeplitz matrix $$\left[\begin{array} \\ a&0&0&0\\ b&a&0&0\\ c&b&a&0\\ d&c&b&a \end{array}\right]$$ using the vector $$\left[\begin{array} .a\\b\\c\\d\end{array}\right]$$ and other binary matrices (i.e. with elements of only $0,1$)?

Doing a Google search for "lower triangular Toeplitz Matrix" comes up with this:

ramanujan.math.trinity.edu/wtrench/research/papers/TRENCH_TN_6.PDF

with the title "Inverses of lower triangular Toeplitz matrices".

Sure, this GNU Octave script does it

T = toeplitz(rand(4,1),[0,0,0,0]);
t = T(:,1); % first column
P = diag(ones(6,1),1) + diag(1,-6); P  = inv(P);
Z = 0*P;
M= [P^0,Z,Z,Z;Z,P^1,Z,Z;Z,Z,P^2,Z;Z,Z,Z,P^3];
v=M*kron(ones(4,1),[zeros(3,1);t]);
T-reshape(v,[7,4])(4:7,:)


The binary matrices used is generator for the matrix representation of cyclic group $C_7$ : ($\bf P$ in the code)

It's exponent placed along the diagonal and fill the rest with zeros. Then pad the $(a,b,c,d)^T$ vector with 3 zeros (or $N-1$ zeros, where $N$ is size of vector)

Trimming can probably be applied to make this smaller and tidier.

The "reshaping" and tidying can be implemented by binary selection matrices.