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%I #5 Mar 31 2012 12:35:45
%S 0,1,1,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1924,1924,0,0,0,0,19799,68302,
%T 19799,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3219407612,
%U 25797991623,25797991623,3219407612,0,0,0,0,0,0
%N T(n,k)=Number of nXk arrays of nonnegative integers with rows and columns in lexicographically increasing order and some row sum or column sum equalling each of the values 1..n+k
%C Table starts
%C .0.1.....0.....0.....0...........0..........0.0.0.0
%C .1.4.....0.....0.....0...........0..........0.0.0
%C .0.0.....0..1924.19799...........0..........0.0
%C .0.0..1924.68302.....0...........0.3219407612
%C .0.0.19799.....0.....0.25797991623
%C .0.0.....0.....0
%C .0.0.....0
%C .0.0
%C .0
%C A solution is possible only if (n+k)*(n+k+1)/2 is even and (n)*(n+1) <= (n+k)*(n+k+1)/2 >= (k)*(k+1)
%e Various solutions
%e ..1......0..1....0..0..1....0..0..0..1....0..0..1....0..0..0..0..1
%e ..2......2..2....0..0..2....0..0..0..2....0..0..2....0..0..0..0..2
%e .................0..1..3....0..0..4..3....0..0..3....0..0..0..0..3
%e .................3..4..0....3..4..1..0....0..3..1....0..0..0..3..3
%e ..........................................5..3..0....0..0..5..5..0
%e .....................................................4..5..2..0..0
%e ...
%e ..0..0..0..1......0..0..0..0..0..1....0..0..0..0..1....0..0..0..1
%e ..0..0..0..2......0..0..0..0..0..2....0..0..0..0..2....0..0..0..2
%e ..0..0..0..3......0..0..0..0..0..3....0..0..0..0..3....0..0..0..3
%e ..0..0..0..4......0..0..0..0..7..3....0..0..0..0..4....0..0..0..4
%e ..0..0..4..1......0..0..3..7..1..0....0..0..0..6..0....0..0..3..2
%e ..0..3..5..0......4..5..3..0..0..0....0..0..8..3..0....0..0..6..0
%e ..6..4..0..0..........................5..7..0..0..0....0..6..1..0
%e .......................................................8..3..0..0
%e ...
%e ..0..0..0..0..0..0..1......0..0..0..0..0..1....0..0..0..0..1
%e ..0..0..0..0..0..0..2......0..0..0..0..0..2....0..0..0..0..2
%e ..0..0..0..0..0..0..3......0..0..0..0..0..3....0..0..0..0..3
%e ..0..0..0..0..0..0..4......0..0..0..0..0..4....0..0..0..0..4
%e ..0..0..0..0..0..6..2......0..0..0..0..0..5....0..0..0..0..5
%e ..0..0..0..0..8..5..0......0..0..0..0..6..0....0..0..0..6..0
%e ..0..0..3..9..2..0..0......0..0..0..7..5..0....0..0..0..7..0
%e ..5..6..4..0..0..0..0......0..1..9..3..0..0....0..0..7..1..0
%e ...........................7..7..0..0..0..0....0..6..5..0..0
%e ...............................................9..4..0..0..0
%K nonn,tabl
%O 1,5
%A _R. H. Hardin_ Jan 03 2011
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