[seqfan] counting r-circulant nXn (0, 1) matrices by row and coum sum

[Richard J. Mathar]

The r-circular (0,1) matrices are not well covered in the OEIS.
A r-circular n X n matrix is a matrix where subsequent rows are obtained
by circular shift of the elements of the previous row by r columns.
See
  P. Zellini, http://dx.doi.org/10.1016/0024-3795(79)90170-8

There is A045655, matching the count for the total number of r-circulant
n X n (0,1) matrices.

Then we might have
 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 4916 3168 2538 992 324 48 1
 1 156 936 3432 8580 15444 20592 20592 15444 8580 3432 936 156 1
which is for n>=0 the number of r-circulant n X n (0,1) matrices where
each row sum and each column sum equals k, where 0<=k<=n is the column
index in the triangle.
Is the column k=1 A002618 ?
For example for n=4 and k=1 we have the 8 matrices
n=4 k= 1 r = 1
1000
0100
0010
0001
--
n=4 k= 1 r = 3
1000
0001
0010
0100
--
n=4 k= 1 r = 1
0100
0010
0001
1000
--
n=4 k= 1 r = 3
0100
1000
0001
0010
--
n=4 k= 1 r = 1
0010
0001
1000
0100
--
n=4 k= 1 r = 3
0010
0100
1000
0001
--
n=4 k= 1 r = 1
0001
1000
0100
0010
--
n=4 k= 1 r = 3
0001
0010
0100
1000

--------------------------
Then we might have
 1
 1 1
 1 2 1
 1 4 4 1
 1 6 6 6 1
 1 6 12 12 6 1
 1 8 18 22 18 8 1
 1 8 24 38 38 24 8 1
 1 16 38 80 86 80 38 16 1
 1 10 40 88 132 132 88 40 10 1
 1 12 50 128 220 262 220 128 50 12 1
 1 12 60 170 340 472 472 340 170 60 12 1
 1 24 92 282 586 936 1050 936 586 282 92 24 1
 1 14 84 292 730 1302 1736 1736 1302 730 292 84 14 1
which is for n>=0 and 0<=k<=n the number of symmetric r-circulant n X n (0,1) matrices
where each row sum and each column sum equals k. Symmetric refers to the
usual matrix algebra where the transpose equals the matrix. 
Is the column k=1 A235384?
For example for n=4 and k=2 we have the following 6 matrices:
n=4 k= 2 r = 3
1100
1001
0011
0110
--
n=4 k= 2 r = 1
1010
0101
1010
0101
--
n=4 k= 2 r = 3
1001
0011
0110
1100
--
n=4 k= 2 r = 3
0110
1100
1001
0011
--
n=4 k= 2 r = 1
0101
1010
0101
1010
--
n=4 k= 2 r = 3
0011
0110
1100
1001
---------------------------------
The we might add for n>=0

0 1
1 2
2 4
3 10
4 20
5 38
6 76
7 142
8 356
9 542
10 1084
11 2110
12 4892
13 8318
which is the number of symmetric r-circulant nXn (0,1) matrices with
constant row and column sums. row sums of the above.
---------------------------------------------------------
Then

 1
 1 1
 1 0 1
 1 3 3 1
 1 0 8 0 1
 1 15 30 30 15 1
 1 0 15 0 15 0 1
 1 35 105 175 175 105 35 1
 1 0 64 0 192 0 64 0 1
 1 27 108 390 378 378 390 108 27 1
 1 0 135 0 570 0 570 0 135 0 1
 1 99 495 1485 2970 4158 4158 2970 1485 495 99 1
 1 0 72 0 762 0 1616 0 762 0 72 0 1
 1 143 858 3146 7865 14157 18876 18876 14157 7865 3146 858 143 1
which is the number of n X n r-circulant (0,1) matrices where each row sum,
each colum sum and the trace is k. For example there are 8 of these
for n=4, k=2:
n=4 k= 2 r = 2
1100
0011
1100
0011
--
n=4 k= 2 r = 3
1100
1001
0011
0110
--
n=4 k= 2 r = 2
1001
0110
1001
0110
--
n=4 k= 2 r = 3
1001
0011
0110
1100
--
n=4 k= 2 r = 2
0110
1001
0110
1001
--
n=4 k= 2 r = 3
0110
1100
1001
0011
--
n=4 k= 2 r = 2
0011
1100
0011
1100
--
n=4 k= 2 r = 3
0011
0110
1100
1001
------------------------------------------------------
Finally there is the number of symmetric nXn r-circulant (0,1) matrices where
each colum sum, each row sum and the trace is k, 0<=k<=n:

 1
 1 1
 1 0 1
 1 3 3 1
 1 0 4 0 1
 1 5 10 10 5 1
 1 0 9 0 9 0 1
 1 7 21 35 35 21 7 1
 1 0 16 0 44 0 16 0 1
 1 9 36 84 126 126 84 36 9 1
 1 0 25 0 100 0 100 0 25 0 1
 1 11 55 165 330 462 462 330 165 55 11 1
 1 0 36 0 246 0 420 0 246 0 36 0 1
 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1
for odd n this is apparently A007318, binomial(n,k).
-----------------------------------------------
In all cases it would be nice to have some independent confirmation
of these counts.

RJM