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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.

%I A334278 #34 Feb 16 2025 08:34:00
%S A334278 0,1,0,-1,1,0,-3,6,-4,1,0,-133,423,-572,441,-214,66,-12,1,0,-3040575,
%T A334278 14412776,-31680240,43389646,-41821924,30276984,-17100952,7701952,
%U A334278 -2794896,818036,-191600,35264,-4936,496,-32,1
%N 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.
%C A334278 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.
%H A334278 Peter Kagey, <a href="/A334278/b334278.txt">Table of n, a(n) for n = 0..68</a> (rows 0..5; row 5 from Andrew Howroyd's file)
%H A334278 Andrew Howroyd, <a href="/A334159/a334159.txt">Chromatic Polynomials of Hypercubes Q_0 to Q_5</a>
%H A334278 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChromaticPolynomial.html">Chromatic Polynomial</a>.
%H A334278 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HypercubeGraph.html">Hypercube Graph</a>.
%F A334278 T(n,0) = 0.
%F A334278 T(n,k) = Sum_{i=1..2^n}, Stirling1(i,k) * A334159(n,i). - _Andrew Howroyd_, Apr 25 2020
%e A334278 Table begins:
%e A334278 n/k| 0     1    2     3    4     5   6    7  8
%e A334278 ---+-------------------------------------------
%e A334278   0| 0,    1
%e A334278   1| 0,   -1,   1
%e A334278   2| 0,   -3,   6,   -4,   1
%e A334278   3| 0, -133, 423, -572, 441, -214, 66, -12, 1
%p A334278 with(GraphTheory): with(SpecialGraphs):
%p A334278 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
%p A334278     ChromaticPolynomial(HypercubeGraph(n), x)):
%p A334278 seq(T(n), n=0..4);  # _Alois P. Heinz_, Jan 14 2025
%t A334278 T[n_, k_] := Coefficient[ChromaticPolynomial[HypercubeGraph[n], x], x, k]
%Y A334278 Cf. A296914 is the reverse of row 3.
%Y A334278 Cf. A334279 is analogous for the n-dimensional cross-polytope, the dual of the n-cube.
%Y A334278 Cf. A001787, A002378, A091940, A140986, A158348.
%Y A334278 Cf. A334159, A334247, A334358.
%K A334278 sign,tabf
%O A334278 0,7
%A A334278 _Peter Kagey_, Apr 21 2020