A252876 T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right n+k-4 and value increasing by 0 or 1 with every step right or down.

0, 0, 0, 1, 1, 1, 3, 8, 8, 3, 6, 26, 44, 26, 6, 10, 61, 153, 153, 61, 10, 15, 120, 413, 615, 413, 120, 15, 21, 211, 949, 1953, 1953, 949, 211, 21, 28, 343, 1948, 5281, 7313, 5281, 1948, 343, 28, 36, 526, 3676, 12686, 23203, 23203, 12686, 3676, 526, 36, 45, 771, 6497, 27805, 64920, 85801, 64920, 27805, 6497, 771, 45

Offset

1,7

Comments

Table starts

..0...0.....1......3......6......10.......15........21........28.........36

..0...1.....8.....26.....61.....120......211.......343.......526........771

..1...8....44....153....413.....949.....1948......3676......6497......10894

..3..26...153....615...1953....5281....12686.....27805.....56624.....108549

..6..61...413...1953...7313...23203....64920....164399....383735.....836797

.10.120...949...5281..23203...85801...277585....806347...2142634....5281314

.15.211..1948..12686..64920..277585..1030330...3407823..10237249...28340232

.21.343..3676..27805.164399..806347..3407823..12742873..42993671..132872804

.28.526..6497..56624.383735.2142634.10237249..42993671.161937617..555632319

.36.771.10894.108549.836797.5281314.28340232.132872804.555632319.2105918045

Links

Alois P. Heinz, Table of n, a(n) for n = 1..2850 (first 479 terms from R. H. Hardin)

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) = (1/2)*n^2 - (3/2)*n + 1

k=2: a(n) = (1/24)*n^4 + (5/12)*n^3 - (13/24)*n^2 - (11/12)*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: with "n+k-3" instead of "n+k-4" T(n,k) = binomial(n+k,k) - 2.

Example

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

Crossrefs

Columns 1-7 give: A000217(n-2), A252870, A252871, A252872, A252873, A252874, A252875.

Main diagonal is A252869.

Cf. A252976, A252930, A323846.

Sequence in context: A301484 A249451 A019744 * A102639 A232182 A118817

Adjacent sequences: A252873 A252874 A252875 * A252877 A252878 A252879

Keyword

nonn,tabl

Author

R. H. Hardin, Dec 24 2014

Status

approved