A253004 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.

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, 140, 279, 69, 69, 279, 140, 35, 56, 285, 1028, 1132, 69, 1132, 1028, 285, 56, 84, 506, 2601, 7235, 3072, 3072, 7235, 2601, 506, 84, 120, 819, 5318, 25233, 39758, 3072

Offset: 1

Author

R. H. Hardin, Dec 25 2014

Keywords

• nonn
• tabl

Examples

			Table starts:
..0...0....0......1.......4.......10........20.........35.........56.........84
..0...0....0......1......14.......55.......140........285........506........819
..0...0....0......1......34......279......1028.......2601.......5318.......9499
..1...1....1......1......69.....1132......7235......25233......63135.....129133
..4..14...34.....69......69.....3072.....39758.....228484.....775433....1932763
.10..55..279...1132....3072.....3072....122833....1486152....8270017...27983105
.20.140.1028...7235...39758...122833....122833....4915726...59154789..329035981
.35.285.2601..25233..228484..1486152...4915726....4915726..204051186.2492354946
.56.506.5318..63135..775433..8270017..59154789..204051186..204051186.8849413857
.84.819.9499.129133.1932763.27983105.329035981.2492354946.8849413857.8849413857
Some solutions for n=6 and k=4:
..0..0..1..2....0..0..1..2....0..0..1..2....0..0..1..1....0..0..1..1
..0..0..1..2....1..1..1..2....0..0..1..2....0..0..1..1....0..1..1..1
..1..1..1..2....1..1..2..2....0..0..1..2....0..1..1..2....1..1..1..1
..1..1..2..2....2..2..2..2....0..1..1..2....1..1..1..2....1..1..1..2
..2..2..2..2....2..2..2..2....1..1..1..2....1..1..1..2....1..2..2..2
..2..2..2..2....2..2..2..2....2..2..2..2....2..2..2..2....2..2..2..2
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..1200
• Robert Dougherty-Bliss, Experimental Methods in Number Theory and Combinatorics, Ph. D. Dissertation, Rutgers Univ. (2024). See p. 21.
• Robert Dougherty-Bliss and Manuel Kauers, Hardinian Arrays, arXiv:2309.00487 [math.CO], 2023. Hardinian Arrays, El. J. Combinat. 31 (2) (2024) #P2.9

Crossrefs

Column 1 is A000292(n-3).

Column 2 is A100157(n-3).

Formula

Empirical for column k:

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

k=2: a(n) = (8/3)*n^3 - 26*n^2 + (253/3)*n - 91 for n>2.

k=3: a(n) = (160/3)*n^3 - 708*n^2 + (9539/3)*n - 4831 for n>4.

k=4: a(n) = (4096/3)*n^3 - 22816*n^2 + (388490/3)*n - 249567 for n>6.

k=5: a(n) = (133120/3)*n^3 - 893616*n^2 + (18332582/3)*n - 14187577 for n>8.

k=6: a(n) = (5242880/3)*n^3 - 41275392*n^2 + (991610656/3)*n - 897487301 for n>10.

k=7: a(n) = (235012096/3)*n^3 - 2126491008*n^2 + (58625640404/3)*n - 60801081325 for n>12.