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%I #20 Apr 25 2024 07:07:20
%S 0,0,0,0,0,0,1,0,0,1,4,1,0,1,4,10,14,1,1,14,10,20,55,34,1,34,55,20,35,
%T 140,279,69,69,279,140,35,56,285,1028,1132,69,1132,1028,285,56,84,506,
%U 2601,7235,3072,3072,7235,2601,506,84,120,819,5318,25233,39758,3072
%N T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 3 and every value within 3 of its king move distance from the upper left and every value increasing by 0 or 1 with every step right, diagonally se or down.
%H R. H. Hardin, <a href="/A253004/b253004.txt">Table of n, a(n) for n = 1..1200</a>
%H Robert Dougherty-Bliss, <a href="https://sites.math.rutgers.edu/~zeilberg/Theses/RobertDoughertyBlissThesis.pdf">Experimental Methods in Number Theory and Combinatorics</a>, Ph. D. Dissertation, Rutgers Univ. (2024). See p. 21.
%H Robert Dougherty-Bliss and Manuel Kauers, <a href="https://arxiv.org/abs/2309.00487">Hardinian Arrays</a>, arXiv:2309.00487 [math.CO], 2023. <a href="https://doi.org/10.37236/12358">Hardinian Arrays</a>, El. J. Combinat. 31 (2) (2024) #P2.9
%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) = (8/3)*n^3 - 26*n^2 + (253/3)*n - 91 for n>2.
%F k=3: a(n) = (160/3)*n^3 - 708*n^2 + (9539/3)*n - 4831 for n>4.
%F k=4: a(n) = (4096/3)*n^3 - 22816*n^2 + (388490/3)*n - 249567 for n>6.
%F k=5: a(n) = (133120/3)*n^3 - 893616*n^2 + (18332582/3)*n - 14187577 for n>8.
%F k=6: a(n) = (5242880/3)*n^3 - 41275392*n^2 + (991610656/3)*n - 897487301 for n>10.
%F k=7: a(n) = (235012096/3)*n^3 - 2126491008*n^2 + (58625640404/3)*n - 60801081325 for n>12.
%e Table starts:
%e ..0...0....0......1.......4.......10........20.........35.........56.........84
%e ..0...0....0......1......14.......55.......140........285........506........819
%e ..0...0....0......1......34......279......1028.......2601.......5318.......9499
%e ..1...1....1......1......69.....1132......7235......25233......63135.....129133
%e ..4..14...34.....69......69.....3072.....39758.....228484.....775433....1932763
%e .10..55..279...1132....3072.....3072....122833....1486152....8270017...27983105
%e .20.140.1028...7235...39758...122833....122833....4915726...59154789..329035981
%e .35.285.2601..25233..228484..1486152...4915726....4915726..204051186.2492354946
%e .56.506.5318..63135..775433..8270017..59154789..204051186..204051186.8849413857
%e .84.819.9499.129133.1932763.27983105.329035981.2492354946.8849413857.8849413857
%e Some solutions for n=6 and k=4:
%e ..0..0..1..2....0..0..1..2....0..0..1..2....0..0..1..1....0..0..1..1
%e ..0..0..1..2....1..1..1..2....0..0..1..2....0..0..1..1....0..1..1..1
%e ..1..1..1..2....1..1..2..2....0..0..1..2....0..1..1..2....1..1..1..1
%e ..1..1..2..2....2..2..2..2....0..1..1..2....1..1..1..2....1..1..1..2
%e ..2..2..2..2....2..2..2..2....1..1..1..2....1..1..1..2....1..2..2..2
%e ..2..2..2..2....2..2..2..2....2..2..2..2....2..2..2..2....2..2..2..2
%Y Column 1 is A000292(n-3).
%Y Column 2 is A100157(n-3).
%K nonn,tabl
%O 1,11
%A _R. H. Hardin_, Dec 25 2014
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