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T(n,k)=Number of nXk arrays containing k copies of 0..n-1 with every element equal to at least one horizontal neighbor and the top left element equal to 0.
6

%I #6 Aug 12 2016 06:55:35

%S 0,1,0,1,1,0,1,1,2,0,1,3,2,6,0,1,5,30,6,24,0,1,12,58,630,24,120,0,1,

%T 26,588,1398,22680,120,720,0,1,60,2388,93672,55896,1247400,720,5040,0,

%U 1,143,17456,609264,33807456,3353880,97297200,5040,40320,0,1,353,102874

%N T(n,k)=Number of nXk arrays containing k copies of 0..n-1 with every element equal to at least one horizontal neighbor and the top left element equal to 0.

%C Table starts

%C .0......1.....1...........1.........1...........1.........1........1.........1

%C .0......1.....1...........3.........5..........12........26.......60.......143

%C .0......2.....2..........30........58.........588......2388....17456....102874

%C .0......6.....6.........630......1398.......93672....609264.20702760.279681294

%C .0.....24....24.......22680.....55896....33807456.331295040

%C .0....120...120.....1247400...3353880.22933860480

%C .0....720...720....97297200.281725200

%C .0...5040..5040.10216206000

%C .0..40320.40320

%C .0.362880

%H R. H. Hardin, <a href="/A267724/b267724.txt">Table of n, a(n) for n = 1..70</a>

%e Some solutions for n=4 k=4

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

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

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

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

%Y Column 2 and column 3 are A000142(n-1).

%Y Column 4 is A007019(n-1).

%K nonn,tabl

%O 1,9

%A _R. H. Hardin_, Jan 20 2016