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T(n,k)=Number of nXk 0..k arrays with every repeated value in every row greater than or equal to, and every column greater than, the previous repeated value.
3

%I #4 Jan 29 2016 07:24:34

%S 2,9,4,64,81,6,615,4096,576,9,7536,378225,216000,3969,12,112476,

%T 56791296,197555340,10941048,25281,16,1981512,12650850576,

%U 371713931400,98956027822,519718464,155236,20,40265487,3926389806144,1257212773492866

%N T(n,k)=Number of nXk 0..k arrays with every repeated value in every row greater than or equal to, and every column greater than, the previous repeated value.

%C Table starts

%C ..2......9..........64............615.............7536................112476

%C ..4.....81........4096.........378225.........56791296...........12650850576

%C ..6....576......216000......197555340.....371713931400......1257212773492866

%C ..9...3969....10941048....98956027822.2335306173137957.120138605995104637373

%C .12..25281...519718464.46876747497685

%C .16.155236.23689358848

%C .20.915849

%C .25

%H R. H. Hardin, <a href="/A268246/b268246.txt">Table of n, a(n) for n = 1..40</a>

%F Empirical for column k:

%F k=1: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4)

%F k=2: [order 15]

%F k=3: [order 40]

%e Some solutions for n=2 k=4

%e ..1..1..0..1....0..0..2..0....0..1..2..0....0..2..2..2....1..1..2..3

%e ..1..0..2..2....3..4..4..3....2..3..1..2....0..3..0..1....1..0..4..2

%Y Column 1 is A002620(n+2).

%Y Column 2 is A267961.

%Y Column 3 is A267971.

%Y Row 1 is A268104.

%Y Row 2 is A268105.

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Jan 29 2016