A253011 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.

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, 231, 632, 124, 124, 632, 231, 35, 56, 489, 2902, 3423, 250, 3423, 2902, 489, 56, 84, 891, 8416, 33533, 14795, 14795, 33533, 8416, 891, 84, 120, 1469, 18770, 158877, 309990

Offset: 1

Author

R. H. Hardin, Dec 25 2014

Keywords

• nonn
• tabl

Comments

Table starts

..0....0.....0.......1........4.........10..........20............35

..0....0.....0.......1.......19.........85.........231...........489

..0....0.....0.......1.......54........632........2902..........8416

..1....1.....1.......1......124.......3423.......33533........158877

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

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

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

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

.56..891.18770..490403.14125312..395069496..9617848524..196919486736

.84.1469.35564.1156178.46481352.2063349297.90094114144.3496191411901

Examples

			Some solutions for n=6 k=4
..0..0..1..1....0..0..0..1....0..0..1..2....0..0..0..1....0..0..0..0
..0..0..1..2....1..1..1..1....1..1..1..2....0..0..0..1....0..0..0..0
..1..1..1..2....1..1..2..2....1..1..1..2....0..0..0..1....0..0..0..1
..1..1..1..2....1..1..2..2....1..1..1..2....0..0..0..1....0..1..1..1
..1..1..1..2....1..2..2..2....1..2..2..2....0..0..1..1....1..1..2..2
..2..2..2..2....2..2..2..2....1..2..2..2....1..1..1..2....1..1..2..2
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..480

Crossrefs

Column 1 is A000292(n-3)

Column 2 is A063496(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) = (16/3)*n^3 - 56*n^2 + (590/3)*n - 231 for n>3

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

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

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

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

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