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.

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

Offset: 0

Author

Kival Ngaokrajang, Sep 29 2015

Keywords

• nonn
• tabf

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.

From Wolfdieter Lang, Oct 12 2015 (Start):

Double symmetry of n X n matrix M: M(i, j) = M(n-i+1, j) = M(i, n-j+1) (= M(n-i+1, n-j+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

Examples

			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.
...
		

Links

• Kival Ngaokrajang, Illustration of initial terms

Crossrefs

Cf. A262666,A002522, A008794, A002416.

Extensions

More terms from Alois P. Heinz, Sep 29 2015