A194480 T(n,k) = number of ways to arrange k indistinguishable points on an n X n X n triangular grid so that no three points are in the same row or diagonal.

1, 0, 3, 0, 3, 6, 0, 1, 15, 10, 0, 0, 17, 45, 15, 0, 0, 6, 105, 105, 21, 0, 0, 0, 114, 410, 210, 28, 0, 0, 0, 39, 879, 1225, 378, 36, 0, 0, 0, 1, 909, 4284, 3066, 630, 45, 0, 0, 0, 0, 337, 8568, 15729, 6762, 990, 55, 0, 0, 0, 0, 15, 8733, 50526, 47565, 13560, 1485, 66, 0, 0, 0

Offset

1,3

Comments

Table starts

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

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

...6...15.....17........6.........0..........0...........0............0

..10...45....105......114........39..........1...........0............0

..15..105....410......879.......909........337..........15............0

..21..210...1225.....4284......8568.......8733........3525..........285

..28..378...3066....15729.....50526......96478.......98473........43713

..36..630...6762....47565....221508.....668028.....1237434......1279905

..45..990..13560...124803....789453....3413828.....9821400.....17860056

..55.1485..25245...293733...2412333...14054915....57367112....159352995

..66.2145..44275...634293...6542316...49171641...268378248...1046727933

..78.3003..73931..1277133..16127397..151422970..1059987987...5488359255

..91.4095.118482..2426424..36762726..420674150..3661533037..24183257037

.105.5460.183365..4389567..78495417.1073422309.11341971885..92740471038

.120.7140.275380..7615062.158548572.2550004472.32090198922.317395080927

.136.9180.402900.12739902.305303544.5699074284.84099053568.987664967535

Links

R. H. Hardin, Table of n, a(n) for n = 1..290

Manuel Kauers and Christoph Koutschan, Some D-finite and some Possibly D-finite Sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023, pp. 31-33.

Formula

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

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

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

T(n,3) = (1/48)*n^6 + (1/16)*n^5 - (3/16)*n^4 + (1/48)*n^3 + (1/6)*n^2 - (1/12)*n

T(n,4) = (1/384)*n^8 + (1/96)*n^7 - (5/64)*n^6 + (13/240)*n^5 + (27/128)*n^4 - (23/96)*n^3 - (13/96)*n^2 + (7/40)*n

T(n,5) = (1/3840)*n^10 + (1/768)*n^9 - (7/384)*n^8 + (37/1920)*n^7 + (737/3840)*n^6 - (2347/3840)*n^5 + (101/192)*n^4 + (93/320)*n^3 - (7/10)*n^2 + (3/10)*n

Example

Some solutions for n=4, k=4:
.....1........0........0........0........0........0........1........1
....1.0......1.0......0.1......0.1......1.0......1.1......0.1......0.1
...0.1.0....1.0.1....0.1.0....0.0.1....0.1.1....0.1.0....1.1.0....0.1.0
..0.0.0.1..0.0.0.1..1.0.1.0..0.1.1.0..0.0.0.1..1.0.0.0..0.0.0.0..0.0.1.0

Crossrefs

Column 1 is A000217.

Column 2 is A050534(n+1).

Sequence in context: A131656 A194492 A194136 * A194485 A120987 A281293

Adjacent sequences: A194477 A194478 A194479 * A194481 A194482 A194483

Keyword

nonn,tabl

Author

R. H. Hardin, Aug 26 2011

Status

approved