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

Author

R. H. Hardin, Dec 24 2014

Keywords

• nonn
• tabl

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

Examples

			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
		

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)

Crossrefs

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

Main diagonal is A252869.

Cf. A252976, A252930, A323846.

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.