A252930 T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right n+k-6 and value increasing by 0 or 1 with every step right or down.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 5, 19, 33, 33, 19, 5, 15, 120, 413, 615, 413, 120, 15, 35, 483, 2859, 6997, 6997, 2859, 483, 35, 70, 1500, 13976, 53950, 84910, 53950, 13976, 1500, 70, 126, 3923, 54199, 315198, 762227, 762227, 315198, 54199, 3923, 126, 210

Offset: 1

Author

R. H. Hardin, Dec 24 2014

Keywords

• nonn
• tabl

Comments

Table starts

...0....0......0........0.........1...........5...........15............35

...0....0......0........1........19.........120..........483..........1500

...0....0......1.......33.......413........2859........13976.........54199

...0....1.....33......615......6997.......53950.......315198.......1499394

...1...19....413.....6997.....84910......762227......5385305......31454256

...5..120...2859....53950....762227.....8241540.....71297441.....512868867

..15..483..13976...315198...5385305....71297441....759337545....6725497344

..35.1500..54199..1499394..31454256...512868867...6725497344...73117894428

..70.3923.177848..6083808.157376166..3160111147..50869309436..675539536773

.126.9069.513905.21733215.692347393.17063990547.335549742230.5411459549576

Examples

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

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)

Crossrefs

Column 1 is A000332(n-1), other columns are A252924 - A252929. Cf. A252923 (diagonal); A252876 (lower right n+k-4), A252976 (lower right n+k-5).

Formula

Empirical for column k:

k=1: a(n) = (1/24)*n^4 - (5/12)*n^3 + (35/24)*n^2 - (25/12)*n + 1.

k=2: [polynomial of degree 8]

k=3: [polynomial of degree 12]

k=4: [polynomial of degree 16]

k=5: [polynomial of degree 20]

k=6: [polynomial of degree 24]

k=7: [polynomial of degree 28]

Empirical: with "n+k-3" instead of "n+k-6" T(n,k) = binomial(n+k,k) - 2.