I was playing with some unrelated topic and randomly stumbled into following:
Suppose we have a $4 \times 4$ Vandermonde matrix where $x_i = (0,1,2,3)$
$ V = \begin{pmatrix} 1 & x_0 & x_0^2 & x_0^3 \\ 1 & x_1 & x_1^2 & x_1^3 \\ 1 & x_2 & x_2^2 & x_2^3 \\ 1 & x_3 & x_3^2 & x_3^3 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \\ 1 & 2 & 4 & 8 \\ 1 & 3 & 9 & 27 \\ \end{pmatrix} $
If we perform LU decomposition of this matrix we get
$V = LU$
$ \begin{pmatrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \\ 1 & 2 & 4 & 8 \\ 1 & 3 & 9 & 27 \\ \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 2 & 1 & 0 \\ 1 & 3 & 3 & 1 \\ \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 2 & 6 \\ 0 & 0 & 0 & 6 \end{pmatrix} $
I noticed that L matrix is composed of binomial coefficients (Pascal's triangle) and U is composed of OEIS A019538 Triangle of numbers sequence (https://oeis.org/A019538).
Hopefully someone can explain why does this weird connection between binomial coefficients and Triangle of numbers sequence appear? Does someone know where could I read/explore more about this topic? Thanks.