A166813 -id:A166813 - cp's OEIS frontend

A252976 T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right n+k-5 and value increasing by 0 or 1 with every step right or down.

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 4, 13, 18, 13, 4, 10, 61, 153, 153, 61, 10, 20, 192, 770, 1236, 770, 192, 20, 35, 483, 2859, 6997, 6997, 2859, 483, 35, 56, 1050, 8694, 30802, 46812, 30802, 8694, 1050, 56, 84, 2058, 22924, 112877, 248182, 248182, 112877, 22924, 2058

Offset: 1

R. H. Hardin, Dec 25 2014

Table starts
..0....0......0.......1........4........10.........20..........35...........56
..0....0......1......13.......61.......192........483........1050.........2058
..0....1.....18.....153......770......2859.......8694.......22924........54272
..1...13....153....1236.....6997.....30802.....112877......359550......1024773
..4...61....770....6997....46812....248182....1100210.....4230324.....14477724
.10..192...2859...30802...248182...1592348....8528422....39423196....161160206
.20..483...8694..112877..1100210...8528422...54926890...303382053...1471499970
.35.1050..22924..359550..4230324..39423196..303382053..1988261908..11360377192
.56.2058..54272.1024773.14477724.161160206.1471499970.11360377192..75922639116
.84.3732.118057.2667554.44951694.593478797.6383377435.57644900961.447545856560

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

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), subtable T_{n X m}(n+m-5).

Cf. A252876, A252930. Column 1 is A000292(n-3). Cf. A252970-A252975 (columns 2-7).

Empirical for column k:
k=1: a(n) = (1/6)*n^3 - 1*n^2 + (11/6)*n - 1
k=2: [polynomial of degree 6]
k=3: [polynomial of degree 9]
k=4: [polynomial of degree 12]
k=5: [polynomial of degree 15]
k=6: [polynomial of degree 18]
k=7: [polynomial of degree 21]
Empirical: with "n+k-3" instead of "n+k-5" T(n,k) = binomial(n+k,k) - 2, see A166810, A166812, A166813.

A166810 Number of n X 6 1..2 arrays containing at least one of each value, all equal values connected, rows considered as a single number in nondecreasing order, and columns considered as a single number in nondecreasing order.

Original entry on oeis.org

5, 26, 82, 208, 460, 922, 1714, 3001, 5003, 8006, 12374, 18562, 27130, 38758, 54262, 74611, 100945, 134594, 177098, 230228, 296008, 376738, 475018, 593773, 736279, 906190, 1107566, 1344902, 1623158, 1947790, 2324782, 2760679, 3262621, 3838378, 4496386, 5245784, 6096452, 7059050, 8145058, 9366817

Offset: 1

R. H. Hardin, Oct 21 2009

nonn

This sequence (and A166812, A166813) correspond to k-tuples x with 0<= x(i+1) <= x(i) <= k except (0,0,0..) and (k,k,k...), where x(i) is the index of the first 2 in row i of the array (or 0 if none); the number of those are the binomials minus 2. - Robert Israel, Nov 23 2015

			Some solutions for n=4
...1.1.1.1.1.2...1.1.1.1.2.2...1.1.1.1.1.1...1.1.1.1.1.1...1.1.1.1.1.1
...1.1.1.1.1.2...1.1.1.1.2.2...1.1.1.1.1.1...1.1.1.1.1.2...1.1.1.1.1.2
...1.1.1.1.2.2...1.1.1.1.2.2...1.1.1.1.1.2...1.1.1.2.2.2...1.1.1.1.1.2
...1.1.1.2.2.2...1.1.1.1.2.2...1.1.1.1.2.2...2.2.2.2.2.2...1.1.1.1.1.2
------
...1.1.1.1.1.2...1.1.1.2.2.2...1.1.1.1.1.2...1.1.1.1.2.2...1.1.1.1.1.2
...1.1.2.2.2.2...1.1.2.2.2.2...1.1.1.1.2.2...1.1.1.1.2.2...1.1.1.1.1.2
...1.2.2.2.2.2...1.1.2.2.2.2...1.1.1.1.2.2...1.2.2.2.2.2...1.1.2.2.2.2
...1.2.2.2.2.2...1.1.2.2.2.2...1.1.2.2.2.2...1.2.2.2.2.2...1.1.2.2.2.2

Robert Israel, Table of n, a(n) for n = 1..10000

seq(binomial(n+6,6)-2, n=1..100); # Robert Israel, Nov 24 2015

Vec(1-2/(1-x)+1/(1-x)^7 + O(x^100)) \\ Altug Alkan, Nov 24 2015

a(n) = A000579(n+6)-2. - R. J. Mathar, Nov 24 2015

G.f.: 1 - 2/(1-x) + 1/(1-x)^7. - Robert Israel, Nov 24 2015

cp's OEIS Frontend

A252976 T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right n+k-5 and value increasing by 0 or 1 with every step right or down.

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