A250812 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction

36, 100, 129, 225, 379, 432, 441, 873, 1315, 1389, 784, 1731, 3081, 4321, 4356, 1296, 3097, 6171, 10233, 13735, 13449, 2025, 5139, 11116, 20631, 32745, 42769, 41112, 3025, 8049, 18537, 37333, 66291, 102393, 131455, 124869, 4356, 12043, 29145, 62469

Offset

1,1

Comments

Table starts

......36.....100.....225......441......784.....1296.....2025......3025

.....129.....379.....873.....1731.....3097.....5139.....8049.....12043

.....432....1315....3081.....6171....11116....18537....29145.....43741

....1389....4321...10233....20631....37333....62469....98481....148123

....4356...13735...32745....66291...120304...201741...318585....479845

...13449...42769..102393...207831...377857...634509..1003089...1512163

...41112..131455..315561...641571..1167796..1962717..3104985...4683421

..124869..400681..963513..1961031..3572173..6007149..9507441..14345803

..377676.1214695.2924265..5955891.10854424.18260061.28908345..43630165

.1139169.3669409.8840313.18013431.32839417.55258029.87498129.132077683

Links

R. H. Hardin, Table of n, a(n) for n = 1..241

Formula

Empirical: T(n,k) = (((5/12)*k^4 + (11/3)*k^3 + (157/12)*k^2 + (95/6)*k + 6)*3^n - ((1/2)*k^4 + (7/2)*k^3 + (23/2)*k^2 + (17/2)*k)*2^n + (1/4)*k^4 + 1*k^3 + (9/4)*k^2 - (1/2)*k)/2

Empirical for column k:

k=1: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (39*3^n-24*2^n+3)/2

k=2: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (126*3^n-99*2^n+20)/2

k=3: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (304*3^n-264*2^n+66)/2

k=4: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (620*3^n-570*2^n+162)/2

k=5: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (1131*3^n-1080*2^n+335)/2

k=6: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (1904*3^n-1869*2^n+618)/2

k=7: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (3016*3^n-3024*2^n+1050)/2

Empirical for row n:

n=1: a(n) = (1/4)*n^4 + (5/2)*n^3 + (37/4)*n^2 + 15*n + 9

n=2: a(n) = 1*n^4 + 10*n^3 + 37*n^2 + 54*n + 27

n=3: a(n) = (15/4)*n^4 + 36*n^3 + (527/4)*n^2 + (359/2)*n + 81

n=4: a(n) = 13*n^4 + 121*n^3 + 439*n^2 + 573*n + 243

n=5: a(n) = (171/4)*n^4 + 390*n^3 + (5627/4)*n^2 + (3575/2)*n + 729

n=6: a(n) = 136*n^4 + 1225*n^3 + 4402*n^2 + 5499*n + 2187

n=7: a(n) = (1695/4)*n^4 + 3786*n^3 + (54287/4)*n^2 + (33539/2)*n + 6561

Example

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

Crossrefs

Row 1 is A000537(n+2)

Sequence in context: A306213 A114819 A137945 * A072413 A131605 A296204

Adjacent sequences: A250809 A250810 A250811 * A250813 A250814 A250815

Keyword

nonn,tabl

Author

R. H. Hardin, Nov 27 2014

Status

approved