A342088 Triangle read by rows: T(n,k) is the number of n-colorings of the vertices of the k-dimensional cross polytope such that no two adjacent vertices have the same color. 0 <= k <= n.

1, 1, 1, 1, 4, 2, 1, 9, 18, 6, 1, 16, 84, 96, 24, 1, 25, 260, 780, 600, 120, 1, 36, 630, 4080, 7560, 4320, 720, 1, 49, 1302, 15330, 61320, 78120, 35280, 5040, 1, 64, 2408, 45696, 351120, 913920, 866880, 322560, 40320

Offset: 0

Author

Peter Kagey, Feb 27 2021

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  n\k| 0   1     2      3       4       5       6       7      8
  ---+----------------------------------------------------------
   0 | 1
   1 | 1,  1
   2 | 1,  4,    2
   3 | 1,  9,   18,     6
   4 | 1, 16,   84,    96,     24
   5 | 1, 25,  260,   780,    600,    120
   6 | 1, 36,  630,  4080,   7560,   4320,    720
   7 | 1, 49, 1302, 15330,  61320,  78120,  35280,   5040
   8 | 1, 64, 2408, 45696, 351120, 913920, 866880, 322560, 40320
		

Links

• Peter Kagey, Rows n = 0..100, flattened
• Eric Weisstein's World of Mathematics, Chromatic Polynomial
• Eric Weisstein's World of Mathematics, Cocktail Party Graph
• Wikipedia, Cross-polytope
• Wikipedia, Turán graph

Crossrefs

Cf. A000012 (k=0), A000290 (k=1), A091940 (k=2), A115400 (k=3), A334281 (k=4), A342073 (k=5), A342074 (k=6), A342075 (k=7).

Cf. A334279.

Programs

Mathematica

T[n_, k_] := Sum[n! k!/((n - k - j)! (k - j)! j!), {j, 0, k}]

Formula

T(n,n) = n!.

T(n,k) = Sum_{i=0..2*k} A334279(k,i)*n^i.

T(n,k) = n*T(n-1,k-1) + n*(n-1)*T(n-2,k-1).

T(n,k) = Sum_{j=0..k} n!k!/((n-k-j)!(k-j)!j!).