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T(n,k)=Number of length n arrays of permutations of 0..n-1 with each element moved by -k to k places and the median of every three consecutive elements nondecreasing.
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%I #4 Oct 22 2015 12:44:42

%S 1,1,2,1,2,3,1,2,6,5,1,2,6,12,8,1,2,6,16,25,13,1,2,6,16,41,57,21,1,2,

%T 6,16,52,108,124,34,1,2,6,16,52,164,280,268,55,1,2,6,16,52,208,476,

%U 729,588,89,1,2,6,16,52,208,676,1428,1908,1285,144,1,2,6,16,52,208,800,2208,4308

%N T(n,k)=Number of length n arrays of permutations of 0..n-1 with each element moved by -k to k places and the median of every three consecutive elements nondecreasing.

%C Table starts

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

%C ..2....2....2.....2.....2.....2.....2.....2.....2.....2.....2.....2.....2.....2

%C ..3....6....6.....6.....6.....6.....6.....6.....6.....6.....6.....6.....6.....6

%C ..5...12...16....16....16....16....16....16....16....16....16....16....16....16

%C ..8...25...41....52....52....52....52....52....52....52....52....52....52....52

%C .13...57..108...164...208...208...208...208...208...208...208...208...208...208

%C .21..124..280...476...676...800...800...800...800...800...800...800...800...800

%C .34..268..729..1428..2208..2900..3360..3360..3360..3360..3360..3360..3360..3360

%C .55..588.1908..4308..7696.10960.14024.16224.16224.16224.16224.16224.16224.16224

%C .89.1285.4969.12816.25508.40792.55492.69212.78088.78088.78088.78088.78088.78088

%H R. H. Hardin, <a href="/A263597/b263597.txt">Table of n, a(n) for n = 1..516</a>

%F Empirical for column k:

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

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

%F k=3: a(n) = a(n-1) +a(n-2) +7*a(n-3) +2*a(n-4) +4*a(n-5) -a(n-7) -a(n-8)

%F k=4: [order 15]

%F k=5: [order 31]

%F k=6: [order 67]

%e Some solutions for n=6 k=4

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

%e ..2....2....0....1....1....5....3....1....5....1....1....0....4....0....1....0

%e ..3....3....2....3....5....2....2....2....0....0....0....1....0....1....3....1

%e ..0....0....4....4....3....3....1....5....2....2....2....5....3....3....5....3

%e ..4....5....3....2....0....4....4....3....4....3....4....3....5....2....2....4

%e ..5....4....5....5....4....1....5....4....3....5....5....4....2....5....4....5

%Y Column 1 is A000045(n+1).

%Y Column 2 is A214663.

%K nonn,tabl

%O 1,3

%A _R. H. Hardin_, Oct 22 2015