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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 up to permutations of rows.
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%I #9 Oct 15 2024 15:42:54

%S 1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,4,2,1,1,1,1,10,6,3,1,1,1,1,26,30,

%T 12,3,1,1,1,1,71,166,117,18,4,1,1,1,1,197,981,1421,345,30,4,1,1,1,1,

%U 554,5937,20326,9691,1042,42,5,1,1,1,1,1570,36646,307063,336596,63076,2746,63,5,1,1

%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 up to permutations of rows.

%C Columns are not permutable.

%C Equivalently, the number of n X k 0..2 arrays with row sums k and column sums n up to permutations of rows.

%H Andrew Howroyd, <a href="/A377063/b377063.txt">Table of n, a(n) for n = 0..434</a> (first 29 antidiagonals)

%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 2 4 10 26 71 197 ...

%e 3 | 1 1 2 6 30 166 981 5937 ...

%e 4 | 1 1 3 12 117 1421 20326 307063 ...

%e 5 | 1 1 3 18 345 9691 336596 12650093 ...

%e 6 | 1 1 4 30 1042 63076 5328136 506525279 ...

%e 7 | 1 1 4 42 2746 369036 76292516 18490880339 ...

%e ...

%Y Main diagonal is A377064.

%Y Rows n=0..4 are A000012, A000012, A257520, A377065, A377066.

%Y Columns k=0..4 are A000012, A000012, A008619, A377067, A377068.

%Y Cf. A334549.

%K nonn,tabl

%O 0,13

%A _Andrew Howroyd_, Oct 14 2024