A334279 Irregular table read by rows: T(n, k) is the coefficient of x^k in the chromatic polynomial of the 1-skeleton of the n-dimensional cross polytope, 0 <= k <= 2n.

0, 0, 1, 0, -3, 6, -4, 1, 0, -64, 154, -137, 58, -12, 1, 0, -2790, 7467, -7852, 4300, -1346, 244, -24, 1, 0, -205056, 593016, -698250, 448015, -175004, 43608, -6990, 700, -40, 1, 0, -22852200, 70164670, -89812001, 64407806, -29113410, 8790285, -1822164, 260868, -25405, 1610, -60, 1

Offset: 1

Author

Peter Kagey, Apr 21 2020

Keywords

• sign
• tabf
• look

Comments

A033815 is the number of acyclic orientations of the n-dimensional cross polytope, which is the absolute value of the chromatic polynomial evaluated at -1.

Sums of absolute values of entries in each row give A033815.

These graphs are chromatically unique, that is, there is no nonisomorphic graph with the same chromatic polynomial.

Conjectures from Peter Kagey, Apr 26 2020: (Start)

T(n,1) = -A161131(2n-1).

T(n,2n-2) = A212689(2n - 1).

T(n,2n-1) = A046092(n-1). (End)

Examples

			Table begins:
n/k| 0       1      2       3      4       5     6     7   8   9 10
---+---------------------------------------------------------------
  1| 0       0      1
  2| 0      -3      6      -4      1
  3| 0     -64    154    -137     58     -12     1
  4| 0   -2790   7467   -7852   4300   -1346   244   -24   1
  5| 0 -205056 593016 -698250 448015 -175004 43608 -6990 700 -40  1
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..2600 (first 50 rows)
• Chong-Yun Chao and George A. Novacky Jr., On maximally saturated graphs, Discrete Math., 41 (1982), 139-143.
• Eric Weisstein's World of Mathematics, Chromatic Polynomial
• Eric Weisstein's World of Mathematics, Cocktail Party Graph
• Wikipedia, Cross-polytope
• Wikipedia, Turán graph

Crossrefs

A334278 is analogous for the n-dimensional hypercube.

Cf. A033815, A115400.

Cf. A161131, A212689, A046092.