A397926 Array read by antidiagonals: T(n,k) is the number of nonnegative integer matrices with a total of n rows and columns with each row sum and column sum nonzero and at most k, n >= 0, k >= 0.

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 3, 2, 2, 0, 1, 0, 4, 6, 13, 0, 0, 1, 0, 5, 12, 47, 36, 6, 0, 1, 0, 6, 20, 122, 266, 201, 0, 0, 1, 0, 7, 30, 260, 1090, 2413, 900, 24, 0, 1, 0, 8, 42, 487, 3300, 15506, 22320, 5562, 0, 0, 1, 0, 9, 56, 833, 8264, 69322, 242702, 266462, 32940, 120, 0

Offset

0,13

Links

Andrew Howroyd, Table of n, a(n) for n = 0..495 (first 31 antidiagonals)

Example

Array begins:
=======================================================
n\k | 0 1   2     3      4       5       6        7 ...
----+--------------------------------------------------
  0 | 1 1   1     1      1       1       1        1 ...
  1 | 0 0   0     0      0       0       0        0 ...
  2 | 0 1   2     3      4       5       6        7 ...
  3 | 0 0   2     6     12      20      30       42 ...
  4 | 0 2  13    47    122     260     487      833 ...
  5 | 0 0  36   266   1090    3300    8264    18144 ...
  6 | 0 6 201  2413  15506   69322  244502   728861 ...
  7 | 0 0 900 22320 242702 1673472 8573280 35533946 ...
  ...
The The T(4,2) = 13 matrices are:
  [1 1]  [0 1]  [1 0]  [0 1]  [2 0]  [0 2]  [1 0]
  [1 1]  [1 0]  [0 1]  [2 0]  [0 1]  [1 0]  [0 2]
.
  [0 2]  [2 0]  [0 1]  [1 0]  [1 1]  [1 1]
  [2 0]  [0 2]  [1 1]  [1 1]  [1 0]  [0 1]

Crossrefs

Main diagonal is A397927.

Rows 0..3 are A000012, A000004, A001477, A002378(n-1).

Columns 0..3 are A000007, A361522, A397928, A397929.

Cf. A257493, A397922 (up to row and column permutations).

Sequence in context: A054876 A397922 A109502 * A323887 A349913 A346236

Adjacent sequences: A397923 A397924 A397925 * A397927 A397929 A397930

Keyword

nonn,tabl,new

Author

Andrew Howroyd, Jul 15 2026

Status

approved