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

1,16

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

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)

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.

Example

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

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).

Sequence in context: A356716 A262700 A243269 * A031019 A324557 A255413

Adjacent sequences: A252927 A252928 A252929 * A252931 A252932 A252933

Keyword

nonn,tabl

Author

R. H. Hardin, Dec 24 2014

Status

approved