A283795 Triangle T(n,k) read by rows: the number of q-circulant n X n {0,1}-matrices where each row sum and each column sum equals k.

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 8, 14, 8, 1, 1, 20, 40, 40, 20, 1, 1, 12, 42, 44, 42, 12, 1, 1, 42, 126, 210, 210, 126, 42, 1, 1, 32, 136, 224, 350, 224, 136, 32, 1, 1, 54, 216, 546, 756, 756, 546, 216, 54, 1, 1, 40, 260, 480, 1200, 1032, 1200, 480, 260, 40, 1, 1, 110, 550, 1650, 3300, 4620, 4620, 3300, 1650, 550, 110, 1, 1, 48, 324, 992, 2538, 3168

Offset

0,5

Comments

q-circulant matrices are constructed by fixing the first row and obtaining the remaining n-1 rows by circularly shifting values by q columns, any q from 0 to n-1.

The triangle is symmetric in each row because flipping 1's and 0's in a matrix gives also a circulant matrix with n-k ones in each row and column.

The number of 1-circulant matrices with k zeros in each row and each column is apparently given by Pascal's Triangle.

Is the column k=1 given by A002618?

Links

Table of n, a(n) for n=0..83.

P. Zellini, On some properties of circulant matrices, Lin. Alg. Applic. 26 (1979) 31-43

Example

The triangle starts in row n=0 and column k=0 as:
1 rsum= 1
1 1 rsum= 2
1 2 1 rsum= 4
1 6 6 1 rsum= 14
1 8 14 8 1 rsum= 32
1 20 40 40 20 1 rsum= 122
1 12 42 44 42 12 1 rsum= 154
1 42 126 210 210 126 42 1 rsum= 758
1 32 136 224 350 224 136 32 1 rsum= 1136
1 54 216 546 756 756 546 216 54 1 rsum= 3146

Crossrefs

Cf. A045655.

Sequence in context: A338874 A338876 A260238 * A168641 A255914 A143185

Adjacent sequences: A283792 A283793 A283794 * A283796 A283797 A283798

Keyword

nonn,tabl

Author

R. J. Mathar, Mar 16 2017

Status

approved