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

1,11

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

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).

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.

Example

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

Crossrefs

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

Sequence in context: A042713 A154130 A041407 * A065564 A332377 A268720

Adjacent sequences: A252973 A252974 A252975 * A252977 A252978 A252979

Keyword

nonn,tabl

Author

R. H. Hardin, Dec 25 2014

Status

approved