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

0,11

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.

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

Links

Table of n, a(n) for n=0..79.

Kival Ngaokrajang, Illustration of initial terms

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

Cf. A262666,A002522, A008794, A002416.

Sequence in context: A340172 A163529 A283655 * A360078 A027354 A192227

Adjacent sequences: A262739 A262740 A262741 * A262743 A262744 A262745

Keyword

nonn,tabf

Author

Kival Ngaokrajang, Sep 29 2015

Extensions

More terms from Alois P. Heinz, Sep 29 2015

Status

approved