Some properties and links concerning M_n (which can be proven strictly by
using the recursive definition of M_n) are:

1) When n is large enough, the number of 1's in (or the Hamming weights of)
the rows of M_n, as well as in the rows of the Sierpinski's triangle, form the
sequence A001316 = 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, ...

2) The number of 0's in the rows of the Sierpinski's triangle form the
sequence A048967 = 0, 0, 1, 0, 3, 2, 3, 0, 7, 6, 7, ...

3) A001316(n)+A048967(n) = A000027(n) = 1, 2, 3, ... - the positive integers.

4) log_2(A001316(n)) = A000120(n) = 0, 1, 1, 2, 1, 2, 2, 3, 1, 2, ... - the
(Hamming) weights of the natural numbers, considered as binary vectors.

5) The decimal numbers corresponding to the binary numbers that the rows of
the Sierpinski's triangle form, are A001317 = 1, 3, 5, 15, 17, 51, 85, 255, ...

6) The transpose matrix M_n^T of M_n represents the relation "precedes"
defined on the n-dimensional Boolean cube as follows: if a=(a_1, a_2, ..., a_n) 
and b=(b_1, b_2, ..., b_n) are n-dimensional binary vectors, a "precedes" b
if a_i <= b_i, for all i=1, 2, ..., n. M_n^T is an important matrix that has many 
other properties related to monotone Boolean functions (see Bakoev V.,
Combinatorial and Algorithmic Properties of One Matrix Structure at Monotone
Boolean Functions, arXiv:1902.06110 [cs.DM], 2019.).