A334549 Array read by antidiagonals: T(n,k) is the number of {-1,0,1} n X k matrices with all rows and columns summing to zero.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 7, 7, 1, 1, 1, 1, 19, 31, 19, 1, 1, 1, 1, 51, 175, 175, 51, 1, 1, 1, 1, 141, 991, 2371, 991, 141, 1, 1, 1, 1, 393, 5881, 32611, 32611, 5881, 393, 1, 1, 1, 1, 1107, 35617, 481381, 1084851, 481381, 35617, 1107, 1, 1

Offset: 0

Author

Andrew Howroyd, May 09 2020

Keywords

• nonn
• tabl

Comments

Equivalently, the number of n X k 0..2 arrays with row sums k and column sums n.

Examples

			Array begins:
====================================================================
n\k | 0 1   2     3       4          5            6            7
----|---------------------------------------------------------------
  0 | 1 1   1     1       1          1            1            1 ...
  1 | 1 1   1     1       1          1            1            1 ...
  2 | 1 1   3     7      19         51          141          393 ...
  3 | 1 1   7    31     175        991         5881        35617 ...
  4 | 1 1  19   175    2371      32611       481381      7343449 ...
  5 | 1 1  51   991   32611    1084851     39612501   1509893001 ...
  6 | 1 1 141  5881  481381   39612501   3680774301 360255871641 ...
  7 | 1 1 393 35617 7343449 1509893001 360255871641 ...
     ...
The T(3,2) = 7 matrices are:
  [0 0]  [ 0  0]  [ 0  0]  [ 1 -1]  [-1  1]  [ 1 -1]  [-1  1]
  [0 0]  [ 1 -1]  [-1  1]  [ 0  0]  [ 0  0]  [-1  1]  [ 1 -1]
  [0 0]  [-1  1]  [ 1 -1]  [-1  1]  [ 1 -1]  [ 0  0]  [ 0  0]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..350 (first 26 antidiagonals)

Crossrefs

Columns k=0..14 are A000012, A000012, A002426, A172634, A172642, A172639, A172633, A172636, A172638, A172641, A172637, A172644, A172640, A172643, A172635.

Main diagonal is A172645.

Cf. A008300, A333901, A376935, A377063 (up to row permutations).

Formula

T(n,k) = T(k,n).