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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 (list; table; graph; refs; listen; history; text; internal format)
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 A296204
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin, Nov 27 2014
STATUS
approved

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Last modified June 14 13:28 EDT 2024. Contains 373400 sequences. (Running on oeis4.)