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%I #7 Jul 17 2026 22:37:18
%S 1,1,0,1,0,0,1,0,1,0,1,0,2,0,0,1,0,3,2,2,0,1,0,4,6,13,0,0,1,0,5,12,47,
%T 36,6,0,1,0,6,20,122,266,201,0,0,1,0,7,30,260,1090,2413,900,24,0,1,0,
%U 8,42,487,3300,15506,22320,5562,0,0,1,0,9,56,833,8264,69322,242702,266462,32940,120,0
%N Array read by antidiagonals: T(n,k) is the number of nonnegative integer matrices with a total of n rows and columns with each row sum and column sum nonzero and at most k, n >= 0, k >= 0.
%H Andrew Howroyd, <a href="/A397926/b397926.txt">Table of n, a(n) for n = 0..495</a> (first 31 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 | 0 0 0 0 0 0 0 0 ...
%e 2 | 0 1 2 3 4 5 6 7 ...
%e 3 | 0 0 2 6 12 20 30 42 ...
%e 4 | 0 2 13 47 122 260 487 833 ...
%e 5 | 0 0 36 266 1090 3300 8264 18144 ...
%e 6 | 0 6 201 2413 15506 69322 244502 728861 ...
%e 7 | 0 0 900 22320 242702 1673472 8573280 35533946 ...
%e ...
%e The The T(4,2) = 13 matrices are:
%e [1 1] [0 1] [1 0] [0 1] [2 0] [0 2] [1 0]
%e [1 1] [1 0] [0 1] [2 0] [0 1] [1 0] [0 2]
%e .
%e [0 2] [2 0] [0 1] [1 0] [1 1] [1 1]
%e [2 0] [0 2] [1 1] [1 1] [1 0] [0 1]
%Y Main diagonal is A397927.
%Y Rows 0..3 are A000012, A000004, A001477, A002378(n-1).
%Y Columns 0..3 are A000007, A361522, A397928, A397929.
%Y Cf. A257493, A397922 (up to row and column permutations).
%K nonn,tabl,new
%O 0,13
%A _Andrew Howroyd_, Jul 15 2026