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%I #9 May 09 2020 19:34:54
%S 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,
%T 175,175,51,1,1,1,1,141,991,2371,991,141,1,1,1,1,393,5881,32611,32611,
%U 5881,393,1,1,1,1,1107,35617,481381,1084851,481381,35617,1107,1,1
%N 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.
%C Equivalently, the number of n X k 0..2 arrays with row sums k and column sums n.
%F T(n,k) = T(k,n).
%e Array begins:
%e ====================================================================
%e n\k | 0 1 2 3 4 5 6 7
%e ----|---------------------------------------------------------------
%e 0 | 1 1 1 1 1 1 1 1 ...
%e 1 | 1 1 1 1 1 1 1 1 ...
%e 2 | 1 1 3 7 19 51 141 393 ...
%e 3 | 1 1 7 31 175 991 5881 35617 ...
%e 4 | 1 1 19 175 2371 32611 481381 7343449 ...
%e 5 | 1 1 51 991 32611 1084851 39612501 1509893001 ...
%e 6 | 1 1 141 5881 481381 39612501 3680774301 360255871641 ...
%e 7 | 1 1 393 35617 7343449 1509893001 360255871641 ...
%e ...
%e The T(3,2) = 7 matrices are:
%e [0 0] [ 0 0] [ 0 0] [ 1 -1] [-1 1] [ 1 -1] [-1 1]
%e [0 0] [ 1 -1] [-1 1] [ 0 0] [ 0 0] [-1 1] [ 1 -1]
%e [0 0] [-1 1] [ 1 -1] [-1 1] [ 1 -1] [ 0 0] [ 0 0]
%Y Columns k=0..14 are A000012, A000012, A002426, A172634, A172642, A172639, A172633, A172636, A172638, A172641, A172637, A172644, A172640, A172643, A172635.
%Y Main diagonal is A172645.
%Y Cf. A008300, A333901.
%K nonn,tabl
%O 0,13
%A _Andrew Howroyd_, May 09 2020
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