A250611 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing x(i,j)-x(i-1,j) in the j direction

11, 26, 26, 57, 64, 57, 120, 140, 140, 120, 247, 290, 297, 290, 247, 502, 586, 592, 592, 586, 502, 1013, 1172, 1153, 1126, 1153, 1172, 1013, 2036, 2336, 2236, 2092, 2092, 2236, 2336, 2036, 4083, 4654, 4353, 3890, 3691, 3890, 4353, 4654, 4083, 8178, 9278

Offset

1,1

Comments

Table starts

...11....26....57...120...247....502...1013...2036...4083...8178..16369...32752

...26....64...140...290...586...1172...2336...4654...9278..18512..36964...73850

...57...140...297...592..1153...2236...4353...8528..16809..33292..66169..131824

..120...290...592..1126..2092...3890...7320..13982..27076..53002.104560..207350

..247...586..1153..2092..3691...6526..11749..21664..40879..78610.153289..301780

..502..1172..2236..3890..6526..10928..18664..32870..59818.112052.214660..417818

.1013..2336..4353..7320.11749..18664..30113..50192..87093.157200.293281..560872

.2036..4654..8528.13982.21664..32870..50192..78814.129104.221798.398368..741758

.4083..9278.16809.27076.40879..59818..87093.129104.198651.321334.548353..982108

.8178.18512.33292.53002.78610.112052.157200.221798.321334.486784.780100.1325186

Links

R. H. Hardin, Table of n, a(n) for n = 1..881

Formula

Empirical: T(n,k) = 2^(n-1)*((k+1)*2)^2 + a quadratic polynomial in n

Empirical for column k (k=2 recurrence also works for k=1):

k=1: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n)=16*2^(n-1) -n-4

k=2: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n)=36*2^(n-1) +n^2-n-10

k=3: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n)=64*2^(n-1) +5*n^2+4*n-16

k=4: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n)=100*2^(n-1) +16*n^2+22*n-18

k=5: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n)=144*2^(n-1) +42*n^2+69*n-8

k=6: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n)=196*2^(n-1) +99*n^2+177*n+30

k=7: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n)=256*2^(n-1) +219*n^2+410*n+128

Empirical for diagonal: a(n) = 11*a(n-1) -52*a(n-2) +138*a(n-3) -225*a(n-4) +231*a(n-5) -146*a(n-6) +52*a(n-7) -8*a(n-8)

Example

Some solutions for n=6 k=4
..1..1..0..0..0....1..1..1..0..0....1..1..1..1..0....0..0..0..1..0
..1..1..0..0..0....1..1..1..1..1....1..1..1..1..0....0..0..0..1..0
..1..1..0..0..0....1..1..1..1..1....1..1..1..1..0....0..0..0..1..0
..0..0..0..0..0....1..1..1..1..1....1..1..1..1..0....0..0..0..1..0
..0..1..1..1..1....1..1..1..1..1....1..1..1..1..0....0..0..0..1..0
..0..1..1..1..1....0..0..0..1..1....1..1..1..1..0....0..0..0..1..0
..0..1..1..1..1....0..0..0..1..1....0..1..1..1..1....0..0..0..1..0

Crossrefs

Column 1 is A000295(n+3)

Sequence in context: A220434 A191720 A066956 * A137015 A260903 A316315

Adjacent sequences: A250608 A250609 A250610 * A250612 A250613 A250614

Keyword

nonn,tabl

Author

R. H. Hardin, Nov 26 2014

Status

approved