

A262742


Irregular table read by rows: T(n,k) is the number of binary symmetric n X n matrices with exactly k 1's; n>=0, 0<=k<=n^2. Where the symmetry axes are in horizontal and vertical.


2



1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 1, 1, 1, 4, 4, 10, 10, 20, 20, 31, 31, 40, 40, 44, 44, 40, 40, 31, 31, 20, 20, 10, 10, 4, 4, 1, 1, 1, 0, 0, 0, 9, 0, 0, 0, 36, 0, 0, 0, 84, 0, 0, 0, 126, 0, 0
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OFFSET

0,11


COMMENTS

The row length of this irregular triangle is n^2+1 = A002522(n).
Inspired by A262666, but rotating the diagonal and antidiagonal symmetry axis to horizontal and vertical axes.
Double symmetry of n X n matrix M: M(i, j) = M(ni+1, j) = M(i, nj+1) (= M(ni+1, nj+1)), here with entries from {0, 1}.
Due to 0 <> 1 flip the rows are symmetric.
The number of independent entries in such an n X n doubly symmetric matrix is A008794(n+1) (squares repeated). Therefore, the row sums give repeated A002416 (omitting the first 1): 1, 2, 2, 16, 16, 512, 512, ... (End)  Wolfdieter Lang, Oct 12 2015


LINKS



EXAMPLE

Irregular table begins:
n\k 0 1 2 3 4 5 6 7 8 9 ...
0: 1
1: 1 1
2: 1 0 0 0 1
3: 1 1 2 2 2 2 2 2 1 1
...
Row 4: 1, 0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 1;
Row 5: 1, 1, 4, 4, 10, 10, 20, 20, 31, 31, 40, 40, 44, 44, 40, 40, 31, 31, 20, 20, 10, 10, 4, 4, 1, 1.
...


CROSSREFS



KEYWORD

nonn,tabf


AUTHOR



EXTENSIONS



STATUS

approved



