A334278 Irregular table read by rows: T(n, k) is the coefficient of x^k in the chromatic polynomial of the cubical graph Q_n, 0 <= k <= 2^n.

0, 1, 0, -1, 1, 0, -3, 6, -4, 1, 0, -133, 423, -572, 441, -214, 66, -12, 1, 0, -3040575, 14412776, -31680240, 43389646, -41821924, 30276984, -17100952, 7701952, -2794896, 818036, -191600, 35264, -4936, 496, -32, 1

Offset: 0

Author

Peter Kagey, Apr 21 2020

Keywords

• sign
• tabf

Comments

The sums of the absolute values of the entries in each row gives A334247, the number of acyclic orientations of edges of the n-cube.

Examples

			Table begins:
n/k| 0     1    2     3    4     5   6    7  8
---+-------------------------------------------
  0| 0,    1
  1| 0,   -1,   1
  2| 0,   -3,   6,   -4,   1
  3| 0, -133, 423, -572, 441, -214, 66, -12, 1
		

Links

• Peter Kagey, Table of n, a(n) for n = 0..68 (rows 0..5; row 5 from Andrew Howroyd's file)
• Andrew Howroyd, Chromatic Polynomials of Hypercubes Q_0 to Q_5
• Eric Weisstein's World of Mathematics, Chromatic Polynomial.
• Eric Weisstein's World of Mathematics, Hypercube Graph.

Crossrefs

Cf. A296914 is the reverse of row 3.

Cf. A334279 is analogous for the n-dimensional cross-polytope, the dual of the n-cube.

Cf. A001787, A002378, A091940, A140986, A158348.

Cf. A334159, A334247, A334358.

Programs

Maple

with(GraphTheory): with(SpecialGraphs):
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
    ChromaticPolynomial(HypercubeGraph(n), x)):
seq(T(n), n=0..4);  # Alois P. Heinz, Jan 14 2025

Mathematica

T[n_, k_] := Coefficient[ChromaticPolynomial[HypercubeGraph[n], x], x, k]

Formula

T(n,0) = 0.

T(n,k) = Sum_{i=1..2^n}, Stirling1(i,k) * A334159(n,i). - Andrew Howroyd, Apr 25 2020