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T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having index change +-(.,.) 0,0 1,2 or 2,2.
10

%I #6 Nov 10 2015 10:52:15

%S 1,2,1,4,5,1,8,25,13,1,16,105,169,34,1,32,441,1573,1156,89,1,64,1869,

%T 14641,20570,7921,233,1,128,7921,146410,366025,269225,54289,610,1,256,

%U 33553,1464100,7320500,9150625,3524125,372100,1597,1,512,142129,14641000

%N T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having index change +-(.,.) 0,0 1,2 or 2,2.

%C Table starts

%C .1....2........4..........8............16...............32..................64

%C .1....5.......25........105...........441.............1869................7921

%C .1...13......169.......1573.........14641...........146410.............1464100

%C .1...34.....1156......20570........366025..........7320500...........146410000

%C .1...89.....7921.....269225.......9150625........398967250.........17394972100

%C .1..233....54289....3524125.....228765625......21860843125.......2089022169025

%C .1..610...372100...46131250....5719140625....1201888840625.....252579343635025

%C .1.1597..2550409..603865625..142978515625...66099082156250...30557658560100100

%C .1.4181.17480761.7904703125.3574462890625.3635200164062500.3696969485250250000

%H R. H. Hardin, <a href="/A264017/b264017.txt">Table of n, a(n) for n = 1..143</a>

%F Empirical for column k:

%F k=1: a(n) = a(n-1)

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

%F k=3: a(n) = 8*a(n-1) -8*a(n-2) +a(n-3)

%F k=4: a(n) = 15*a(n-1) -25*a(n-2) for n>4

%F k=5: a(n) = 25*a(n-1) for n>3

%F k=6: a(n) = 60*a(n-1) -300*a(n-2) +1500*a(n-3) -7500*a(n-4) +3125*a(n-5) for n>7

%F k=7: [order 13] for n>15

%F Empirical for row n:

%F n=1: a(n) = 2*a(n-1)

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

%F n=3: a(n) = 10*a(n-1) for n>5

%F n=4: a(n) = 19*a(n-1) +304*a(n-3) +256*a(n-4) for n>8

%F n=5: [order 14] for n>18

%F n=6: [order 25] for n>31

%e Some solutions for n=4 k=4

%e ..0..8..9..3..4....0.13..2..3..4....7..1.14..3..4....7..1.14..3..4

%e ..5.13.14..1..2...12..6..7..8..9...17..6..0..8..9...12.13..0..8..9

%e .17.18.12..6..7...22.23.24..1.14...10.11..5.13..2...17.18..5..6..2

%e .15.16.10.11.19...15.16..5.18.19...15.23.24.18.19...15.16.10.11.19

%e .20.21.22.23.24...20.21.10.11.17...20.21.22.16.12...20.21.22.23.24

%Y Column 2 is A001519(n+1).

%Y Column 3 is A081068.

%Y Row 1 is A000079(n-1).

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Nov 01 2015