[seqfan] Problems posed regarding A004736 and A003983.

[Ed Jeffery]

The two sequences are A004736 <https://oeis.org/A004736> and
A003983<https://oeis.org/A003983>.
First, A004736 is the triangle

1;
2,1;
3,2,1;
4,3,2,1;
5,4,3,2,1;
...,

in which the pattern for the n-th row continues as {n, n-1, ..., 2, 1}. Can
someone prove that this triangle, taken as an infinite lower-triangular
matrix, is equal to B^2, where

B=[1,0,...; 1,1,0,...; 1,1,1,0,...; ...] (here 0,... means 0,0,0,...)?

Second, A003983 is the triangle

1;
1,1;
1,2,1;
1,2,2,1;
1,2,3,2,1;
...,

in which you "...count up to ceiling(n/2) and back down again (repeating
the central term when n is even)," as described by Franklin T.
Adams-Watters.

I was trying to find a lower-triangular matrix whose square gives this
triangle but failed. I also got no clues from the generating function.
However, arranging the diagonals as rows and letting the sequence be read
from the antidiagonals of the resulting infinite (square) array

A=
1,1,1,1,1,...;
1,2,2,2,2,...;
1,2,3,3,3,...;
1,2,3,4,4,...;
1,2,3,4,5,...;
...,

can someone prove that A = M^2, where M is the infinite square matrix (if
that makes sense)

M = [0,...,0,1; 0,...,0,1,1; 0,...,0,1,1,1; ...]?

Finally, can someone prove that

B*M = [0,...,0,1; 0,...,0,1,2; 0,...,0,1,2,3; ...]?


LEJ