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
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
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.
Comments