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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.
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%I #15 Oct 15 2024 15:42:24

%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.

%H Andrew Howroyd, <a href="/A334549/b334549.txt">Table of n, a(n) for n = 0..350</a> (first 26 antidiagonals)

%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, A376935, A377063 (up to row permutations).

%K nonn,tabl

%O 0,13

%A _Andrew Howroyd_, May 09 2020