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

Author

R. H. Hardin, Aug 26 2011

Keywords

• nonn
• tabl

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

Examples

			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
		

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.

Crossrefs

Column 1 is A000217.

Column 2 is A050534(n+1).

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