A262666 Irregular table read by rows: T(n,k) is the number of binary bisymmetric n X n matrices with exactly k 1's; n>=0, 0<=k<=n^2.

1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 0, 4, 0, 8, 0, 12, 0, 14, 0, 12, 0, 8, 0, 4, 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, 6, 0, 21, 0, 56, 0, 120, 0, 216, 0, 336, 0, 456, 0

Offset

0,6

Comments

T(n,k) = 0 if n is even and k is odd.

T(n,k) = T(n,k+1) if n is odd and k is even.

Links

Alois P. Heinz, Rows n = 0..32, flattened

Kival Ngaokrajang, Illustration of initial terms

Dennis P. Walsh, Notes on binary bisymmetric matrices

Wikipedia, Bisymmetric Matrix

Formula

G.f. for row n: (1+x)^t*(1+x^2)^(n-t)*(1+x^4)^(((n-2)*n+t)/4) where t = n mod 2. - Alois P. Heinz, Sep 27 2015

Example

Irregular table begins:
n\k 0   1   2   3   4   5   6   7   8   9   ...
0:  1
1:  1   1
2:  1   0   2   0   1
3:  1   1   2   2   2   2   2   2   1   1
4:  1   0   4   0   8   0  12   0  14   0   ...
5:  1   1   4   4  10  10  20  20  31  31   ...
...

Maple

T:= n-> seq(coeff((t->(1+x^2)^(n-t)*(1+x)^t*(1+x^4)^
      (((n-2)*n+t)/4))(irem(n, 2)), x, i), i=0..n^2):
seq(T(n), n=0..6);  # Alois P. Heinz, Sep 27 2015

Crossrefs

Row sums give A060656(n+1).

Columns k=0-3 give: A000012, A000035, A052928, A237420(n+1).

Cf. A184948, A261242.

Sequence in context: A349465 A035697 A135549 * A124737 A121303 A166396

Adjacent sequences: A262663 A262664 A262665 * A262667 A262668 A262669

Keyword

nonn,tabf

Author

Kival Ngaokrajang, Sep 26 2015

Extensions

More terms from Alois P. Heinz, Sep 27 2015

Status

approved