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

Author

R. H. Hardin, Nov 27 2014

Keywords

• nonn
• tabl

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

Examples

			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
		

Links

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

Crossrefs

Row 1 is A000537(n+2)

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