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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.

%I A334549 #15 Oct 15 2024 15:42:24
%S A334549 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 A334549 175,175,51,1,1,1,1,141,991,2371,991,141,1,1,1,1,393,5881,32611,32611,
%U A334549 5881,393,1,1,1,1,1107,35617,481381,1084851,481381,35617,1107,1,1
%N 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.
%C A334549 Equivalently, the number of n X k 0..2 arrays with row sums k and column sums n.
%H A334549 Andrew Howroyd, <a href="/A334549/b334549.txt">Table of n, a(n) for n = 0..350</a> (first 26 antidiagonals)
%F A334549 T(n,k) = T(k,n).
%e A334549 Array begins:
%e A334549 ====================================================================
%e A334549 n\k | 0 1   2     3       4          5            6            7
%e A334549 ----|---------------------------------------------------------------
%e A334549   0 | 1 1   1     1       1          1            1            1 ...
%e A334549   1 | 1 1   1     1       1          1            1            1 ...
%e A334549   2 | 1 1   3     7      19         51          141          393 ...
%e A334549   3 | 1 1   7    31     175        991         5881        35617 ...
%e A334549   4 | 1 1  19   175    2371      32611       481381      7343449 ...
%e A334549   5 | 1 1  51   991   32611    1084851     39612501   1509893001 ...
%e A334549   6 | 1 1 141  5881  481381   39612501   3680774301 360255871641 ...
%e A334549   7 | 1 1 393 35617 7343449 1509893001 360255871641 ...
%e A334549      ...
%e A334549 The T(3,2) = 7 matrices are:
%e A334549   [0 0]  [ 0  0]  [ 0  0]  [ 1 -1]  [-1  1]  [ 1 -1]  [-1  1]
%e A334549   [0 0]  [ 1 -1]  [-1  1]  [ 0  0]  [ 0  0]  [-1  1]  [ 1 -1]
%e A334549   [0 0]  [-1  1]  [ 1 -1]  [-1  1]  [ 1 -1]  [ 0  0]  [ 0  0]
%Y A334549 Columns k=0..14 are A000012, A000012, A002426, A172634, A172642, A172639, A172633, A172636, A172638, A172641, A172637, A172644, A172640, A172643, A172635.
%Y A334549 Main diagonal is A172645.
%Y A334549 Cf. A008300, A333901, A376935, A377063 (up to row permutations).
%K A334549 nonn,tabl
%O A334549 0,13
%A A334549 _Andrew Howroyd_, May 09 2020