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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
15
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
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 A376119
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin, Nov 27 2014
STATUS
approved