A194492 T(n,k)=Number of ways to arrange k nonattacking knights on the lower triangle of an n X n board

1, 0, 3, 0, 3, 6, 0, 1, 13, 10, 0, 0, 12, 37, 15, 0, 0, 4, 62, 87, 21, 0, 0, 0, 46, 253, 178, 28, 0, 0, 0, 13, 407, 804, 328, 36, 0, 0, 0, 1, 387, 2168, 2136, 558, 45, 0, 0, 0, 0, 230, 3727, 8685, 4958, 892, 55, 0, 0, 0, 0, 88, 4257, 23496, 28376, 10376, 1357, 66, 0, 0, 0, 0, 20

Offset

1,3

Comments

Table starts

...1....0......0........0.........0..........0...........0............0

...3....3......1........0.........0..........0...........0............0

...6...13.....12........4.........0..........0...........0............0

..10...37.....62.......46........13..........1...........0............0

..15...87....253......407.......387........230..........88...........20

..21..178....804.....2168......3727.......4257........3300.........1739

..28..328...2136.....8685.....23496......44005.......58630........56795

..36..558...4958....28376....111433.....312296......641678.......986879

..45..892..10376....79611....429343....1693828.....5025711.....11453482

..55.1357..20013...198334...1407755....7449231....30209361.....95826217

..66.1983..36144...449336...4061432...27785786...147914590....625250362

..78.2803..61846...942072..10561723...90732814...614046090...3340670026

..91.3853.101163..1852096..25208796..265594944..2226985986..15164906168

.105.5172.159286..3449261..55996244..709634275..7218141771..60136468319

.120.6802.242748..6133944.117021864.1755164932.21280624486.212856667131

.136.8788.359634.10482661.232070481.4063548824.57867863073.683992163441

Links

R. H. Hardin, Table of n, a(n) for n = 1..421

Formula

Empirical: T(n,k) is a polynomial of degree 2k in n for n>3*k-5, with lead coefficient 1/(2^k*k!)

T(n,1) = (1/2)*n^2 + (1/2)*n

T(n,2) = (1/8)*n^4 + (1/4)*n^3 - (17/8)*n^2 + (31/4)*n - 8 for n>1

T(n,3) = (1/48)*n^6 + (1/16)*n^5 - (17/16)*n^4 + (133/48)*n^3 + (433/24)*n^2 - (743/6)*n + 218 for n>4

T(n,4) = (1/384)*n^8 + (1/96)*n^7 - (17/64)*n^6 + (5/12)*n^5 + (1571/128)*n^4 - (6653/96)*n^3 - (8641/96)*n^2 + (13951/8)*n - 4161 for n>7

T(n,5) = (1/3840)*n^10 + (1/768)*n^9 - (17/384)*n^8 + (3/128)*n^7 + (951/256)*n^6 - (68939/3840)*n^5 - (1433/12)*n^4 + (230047/192)*n^3 - (149581/240)*n^2 - (718267/30)*n + 72425 for n>10

T(n,6) = (1/46080)*n^12 + (1/7680)*n^11 - (17/3072)*n^10 - (13/4608)*n^9 + (6635/9216)*n^8 - (66667/23040)*n^7 - (2096789/46080)*n^6 + (1744105/4608)*n^5 + (10251671/11520)*n^4 - (107575289/5760)*n^3 + (46056797/1440)*n^2 + (984185/3)*n - 1221248 for n>13

T(n,7) = (1/645120)*n^14 + (1/92160)*n^13 - (17/30720)*n^12 - (79/92160)*n^11 + (1047/10240)*n^10 - (29857/92160)*n^9 - (946249/92160)*n^8 + (48326573/645120)*n^7 + (7473253/15360)*n^6 - (153662159/23040)*n^5 - (1018307/768)*n^4 + (40217689/144)*n^3 - (212485407/280)*n^2 - (316047321/70)*n + 20362649 for n>16

Example

Some solutions for n=5 k=4
..0..........0..........0..........0..........0..........1..........0
..0.1........0.1........0.0........0.0........0.1........0.0........1.0
..0.0.0......1.1.1......1.1.0......0.0.0......0.0.0......0.0.0......0.0.0
..0.0.0.0....0.0.0.0....1.1.0.0....0.1.0.0....0.0.0.0....1.1.0.0....0.0.0.1
..0.1.0.1.1..0.0.0.0.0..0.0.0.0.0..0.1.1.0.1..1.1.1.0.0..0.0.0.0.1..1.0.1.0.0

Crossrefs

Column 1 is A000217

Sequence in context: A372865 A014715 A131656 * A194136 A194480 A194485

Adjacent sequences: A194489 A194490 A194491 * A194493 A194494 A194495

Keyword

nonn,tabl

Author

R. H. Hardin Aug 26 2011

Status

approved