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T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 3 and every value increasing by 0 or 1 with every step right, diagonally se or down
7

%I #4 Dec 25 2014 19:31:55

%S 0,0,0,0,0,0,1,0,0,1,4,1,0,1,4,10,19,1,1,19,10,20,85,54,1,54,85,20,35,

%T 231,632,124,124,632,231,35,56,489,2902,3423,250,3423,2902,489,56,84,

%U 891,8416,33533,14795,14795,33533,8416,891,84,120,1469,18770,158877,309990

%N T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 3 and every value increasing by 0 or 1 with every step right, diagonally se or down

%C Table starts

%C ..0....0.....0.......1........4.........10..........20............35

%C ..0....0.....0.......1.......19.........85.........231...........489

%C ..0....0.....0.......1.......54........632........2902..........8416

%C ..1....1.....1.......1......124.......3423.......33533........158877

%C ..4...19....54.....124......250......14795......309990.......2853292

%C .10...85...632....3423....14795......54219.....2327062......43197859

%C .20..231..2902...33533...309990....2327062....14697256.....541365583

%C .35..489..8416..158877..2853292...43197859...541365583....5713126349

%C .56..891.18770..490403.14125312..395069496..9617848524..196919486736

%C .84.1469.35564.1156178.46481352.2063349297.90094114144.3496191411901

%H R. H. Hardin, <a href="/A253011/b253011.txt">Table of n, a(n) for n = 1..480</a>

%F Empirical for column k:

%F k=1: a(n) = (1/6)*n^3 - 1*n^2 + (11/6)*n - 1

%F k=2: a(n) = (16/3)*n^3 - 56*n^2 + (590/3)*n - 231 for n>3

%F k=3: a(n) = (800/3)*n^3 - 3980*n^2 + (60442/3)*n - 34576 for n>6

%F k=4: a(n) = (65536/3)*n^3 - 422912*n^2 + (8343128/3)*n - 6208382 for n>9

%F k=5: a(n) = 2973696*n^3 - 70716096*n^2 + 570494664*n - 1560617160 for n>12

%F k=6: [polynomial of degree 3] for n>15

%F k=7: [polynomial of degree 3] for n>18

%e Some solutions for n=6 k=4

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

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

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

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

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

%e ..2..2..2..2....2..2..2..2....1..2..2..2....1..1..1..2....1..1..2..2

%Y Column 1 is A000292(n-3)

%Y Column 2 is A063496(n-3)

%K nonn,tabl

%O 1,11

%A _R. H. Hardin_, Dec 25 2014