A252820 T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and every value within 2 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, 7, 17, 17, 7, 11, 40, 63, 40, 11, 16, 81, 187, 187, 81, 16, 22, 147, 468, 684, 468, 147, 22, 29, 246, 1032, 2078, 2078, 1032, 246, 29, 37, 387, 2067, 5490, 7564, 5490, 2067, 387, 37, 46, 580, 3840, 13015, 23664, 23664, 13015, 3840, 580, 46, 56, 836, 6716

Offset: 1

Author

R. H. Hardin, Dec 22 2014

Keywords

• nonn
• tabl

Comments

Table starts

..1...2.....4......7.....11......16.......22........29........37.........46

..2...6....17.....40.....81.....147......246.......387.......580........836

..4..17....63....187....468....1032.....2067......3840......6716......11179

..7..40...187....684...2078....5490....13015.....28299.....57338.....109549

.11..81...468...2078...7564...23664....65711....165685....385736.....839799

.16.147..1032...5490..23664...86724...279300....809349...2147638....5289321

.22.246..2067..13015..65711..279300..1033761...3414257..10248688...28359679

.29.387..3840..28299.165685..809349..3414257..12755742..43017980..132916561

.37.580..6716..57338.385736.2147638.10248688..43017980.161986236..555724696

.46.836.11179.109549.839799.5289321.28359679.132916561.555724696.2106102800

Examples

			Some solutions for n=4 k=4
..0..1..2..2....0..1..2..3....0..0..1..1....0..1..2..3....0..0..0..1
..1..1..2..3....1..1..2..3....0..1..2..2....1..2..3..3....0..1..1..2
..2..2..3..4....1..1..2..3....1..2..3..3....2..3..3..4....1..2..2..3
..2..3..4..5....1..2..3..4....1..2..3..4....2..3..4..4....2..2..3..4
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..1680

Crossrefs

Column 1 is A000124(n-1)

Formula

Empirical for column k:

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

k=2: a(n) = (1/24)*n^4 + (5/12)*n^3 - (1/24)*n^2 + (7/12)*n + 1

k=3: [polynomial of degree 6]

k=4: [polynomial of degree 8]

k=5: [polynomial of degree 10]

k=6: [polynomial of degree 12]

k=7: [polynomial of degree 14]

Empirical for "within 1" instead of "within 2" is T(n,k)=binomial(n+k,k)-1