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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.

%I A334279 #41 Feb 16 2025 08:34:00
%S A334279 0,0,1,0,-3,6,-4,1,0,-64,154,-137,58,-12,1,0,-2790,7467,-7852,4300,
%T A334279 -1346,244,-24,1,0,-205056,593016,-698250,448015,-175004,43608,-6990,
%U A334279 700,-40,1,0,-22852200,70164670,-89812001,64407806,-29113410,8790285,-1822164,260868,-25405,1610,-60,1
%N 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.
%C A334279 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.
%C A334279 Sums of absolute values of entries in each row give A033815.
%C A334279 These graphs are chromatically unique, that is, there is no nonisomorphic graph with the same chromatic polynomial.
%C A334279 Conjectures from _Peter Kagey_, Apr 26 2020: (Start)
%C A334279 T(n,1) = -A161131(2n-1).
%C A334279 T(n,2n-2) = A212689(2n - 1).
%C A334279 T(n,2n-1) = A046092(n-1). (End)
%H A334279 Peter Kagey, <a href="/A334279/b334279.txt">Table of n, a(n) for n = 1..2600</a> (first 50 rows)
%H A334279 Chong-Yun Chao and George A. Novacky Jr., <a href="https://doi.org/10.1016/0012-365X(82)90200-X">On maximally saturated graphs</a>, Discrete Math., 41 (1982), 139-143.
%H A334279 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChromaticPolynomial.html">Chromatic Polynomial</a>
%H A334279 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>
%H A334279 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cross-polytope">Cross-polytope</a>
%H A334279 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tur%C3%A1n_graph">Turán graph</a>
%e A334279 Table begins:
%e A334279 n/k| 0       1      2       3      4       5     6     7   8   9 10
%e A334279 ---+---------------------------------------------------------------
%e A334279   1| 0       0      1
%e A334279   2| 0      -3      6      -4      1
%e A334279   3| 0     -64    154    -137     58     -12     1
%e A334279   4| 0   -2790   7467   -7852   4300   -1346   244   -24   1
%e A334279   5| 0 -205056 593016 -698250 448015 -175004 43608 -6990 700 -40  1
%Y A334279 A334278 is analogous for the n-dimensional hypercube.
%Y A334279 Cf. A033815, A115400.
%Y A334279 Cf. A161131, A212689, A046092.
%K A334279 sign,tabf,look
%O A334279 1,5
%A A334279 _Peter Kagey_, Apr 21 2020