A252976 T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right n+k-5 and value increasing by 0 or 1 with every step right or down.

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 4, 13, 18, 13, 4, 10, 61, 153, 153, 61, 10, 20, 192, 770, 1236, 770, 192, 20, 35, 483, 2859, 6997, 6997, 2859, 483, 35, 56, 1050, 8694, 30802, 46812, 30802, 8694, 1050, 56, 84, 2058, 22924, 112877, 248182, 248182, 112877, 22924, 2058

Offset: 1

Author

R. H. Hardin, Dec 25 2014

Keywords

• nonn
• tabl

Comments

Table starts

..0....0......0.......1........4........10.........20..........35...........56

..0....0......1......13.......61.......192........483........1050.........2058

..0....1.....18.....153......770......2859.......8694.......22924........54272

..1...13....153....1236.....6997.....30802.....112877......359550......1024773

..4...61....770....6997....46812....248182....1100210.....4230324.....14477724

.10..192...2859...30802...248182...1592348....8528422....39423196....161160206

.20..483...8694..112877..1100210...8528422...54926890...303382053...1471499970

.35.1050..22924..359550..4230324..39423196..303382053..1988261908..11360377192

.56.2058..54272.1024773.14477724.161160206.1471499970.11360377192..75922639116

.84.3732.118057.2667554.44951694.593478797.6383377435.57644900961.447545856560

Examples

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

Links

• R. H. Hardin, Table of n, a(n) for n = 1..449
• R. J. Mathar, Counting 2-way monotonic terrace forms over rectangular landscapes, vixra 1511.0225 (2015), subtable T_{n X m}(n+m-5).

Crossrefs

Cf. A252876, A252930. Column 1 is A000292(n-3). Cf. A252970-A252975 (columns 2-7).

Formula

Empirical for column k:

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

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: with "n+k-3" instead of "n+k-5" T(n,k) = binomial(n+k,k) - 2, see A166810, A166812, A166813.