A252828 T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and every value within 3 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.

1, 2, 2, 4, 6, 4, 8, 18, 18, 8, 15, 53, 81, 53, 15, 26, 142, 340, 340, 142, 26, 42, 339, 1238, 1920, 1238, 339, 42, 64, 729, 3891, 9075, 9075, 3891, 729, 64, 93, 1437, 10761, 36292, 54376, 36292, 10761, 1437, 93, 130, 2638, 26764, 125892, 271846, 271846, 125892

Offset: 1

Author

R. H. Hardin, Dec 22 2014

Keywords

• nonn
• tabl

Comments

Table starts

...1....2......4.......8.......15........26.........42..........64...........93

...2....6.....18......53......142.......339........729........1437.........2638

...4...18.....81.....340.....1238......3891......10761.......26764........60988

...8...53....340....1920.....9075.....36292.....125892......387849......1082111

..15..142...1238....9075....54376....271846....1165921.....4396009.....14863460

..26..339...3891...36292...271846...1679072....8807722....40232545....163307844

..42..729..10761..125892..1165921...8807722...55960651...306796310...1481748658

..64.1437..26764..387849..4396009..40232545..306796310..2001017650..11403395172

..93.2638..60988.1082111.14863460.163307844.1481748658.11403395172..76084625352

.130.4568.129236.2777103.45791493.598768118.6411737114.57777817522.448101581256

Examples

			Some solutions for n=4 k=4
..0..0..0..1....0..1..2..3....0..0..1..2....0..0..1..1....0..1..2..2
..0..1..1..2....1..2..2..3....1..1..2..2....0..1..1..1....1..1..2..2
..0..1..2..3....2..3..3..4....1..2..2..3....0..1..2..2....1..1..2..2
..1..2..3..4....3..3..3..4....2..2..2..3....1..2..2..3....1..2..2..3
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..721

Crossrefs

Column 1 is A000125(n-1)

Formula

Empirical for column k:

k=1: a(n) = (1/6)*n^3 - (1/2)*n^2 + (4/3)*n

k=2: [polynomial of degree 6]

k=3: [polynomial of degree 9]

k=4: [polynomial of degree 12]

k=5: [polynomial of degree 15]

k=6: [polynomial of degree 18]

k=7: [polynomial of degree 21]

Empirical for "within 1" instead of "within 3" is T(n,k)=binomial(n+k,k)-1