
COMMENTS

The row length sequence is 1 for n = 1 and A000982(n2) + 1 for n >= 2, that is: 1, 1, 2, 3, 6, 9, 14, 19, 26, 33, 42, ... = A261243.
This entry is motivated by A258643.
For bisymmetric matrices see the Wikipedia link.
For the number of independent entries of an n X n bisymmetric matrix B_n see a Jul 07 2015 comment on A002620(n+1), n >= 1. For the binary case (only 0 and 1 entries) see A060656(n+1), and the Dennis P. Walsh comment and link. If B_n[1,1] and B_n[1,n] is given then the four corners are fixed, and, for n >= 3, there are A002620(n+1)  2 = A014616(n2) entries free.
If the n X n bisymmetric matrix B_n of 0s and 1s with B_n[1, 1] = 1 = B_n[1, n] is considered as a grid of n^2 squares of length 1 (in some length unit) with the four corners filled with 1s and the other squares with 0 or 1 then a path between the centers of squares with step length 1 can be defined. No diagonal steps (length sqrt(2)) are allowed. B_n is called connected if there exists no path of 0s which dissects the grid into two parts.
An island of 0s (a 0island) in B_n is defined as a set of 0s for which each pair is connected by a path of 0s, and a 0 entry at the coast of a 0island has at least one entry 1 one step away. A single square filled with a 0 is a 0island if all four neighbors 1 step (of length 1)apart are filled with 1s. If k=0 there exists no such 0island. See the n=4 examples with k >=1 below. The k = 1 matrix has one simply connected 0island of four squares. The four k = 2 matrices have two 0islands consisting of one square each.
See the link with the figures by K. N. where red squares stand for 1 and empty squares for 0. Each matrix appears there rotated by 45 degrees in the counterclockwise direction. The mirror operation means row reversion in the matrix B_n. In the figures this is a mirror operation w.r.t. the middle NWSE diagonal. 0islands appear in the figures as holes.
For the row sums see A261244.


EXAMPLE

The irregular triangle T(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 ...
1: 1
2: 1
3: 2 1
4: 4 1 4
5: 12 18 12 8 6 2
6: 44 56 120 28 88 4 36 0 8
...
n=4: k=0:
[[1,1,1,1], [1,1,1,1], [1,1,1,1], [1,1,1,1]],
[[1,0,0,1], [0,1,1,0], [0,1,1,0], [1,0,0,1]],
[[1,1,0,1], [1,1,1,0], [0,1,1,1], [1,0,1,1]],
[[1,0,1,1], [0,1,1,1], [1,1,1,0], [1,1,0,1]];
k=1:
[[1,1,1,1], [1,0,0,1], [1,0,0,1], [1,1,1,1]];
k=2:
[[1,1,1,1], [1,0,1,1], [1,1,0,1], [1,1,1,1]],
[[1,1,1,1], [1,1,0,1], [1,0,1,1], [1,1,1,1]],
[[1,1,0,1], [1,0,1,0], [0,1,0,1], [1,0,1,1]],
[[1,0,1,1], [0,1,0,1], [1,0,1,0], [1,1,0,1]].
