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%I #36 Aug 21 2020 11:07:29
%S 0,0,-1,1,0,2,-3,1,0,-3,6,-4,1,0,4,-10,10,-5,1,0,-5,15,-20,15,-6,1,0,
%T 6,-21,35,-35,21,-7,1,0,-7,28,-56,70,-56,28,-8,1,0,8,-36,84,-126,126,
%U -84,36,-9,1,0,-9,45,-120,210,-252,210,-120,45,-10,1,0,10
%N Triangle read by rows: row n gives the coefficients in the expansion of the chromatic polynomial of the n-cycle graphs.
%C The chromatic polynomial of an n-cycle graph is p(x;n) = (x - 1)^n + (-1)^n*(x - 1). - _Franck Maminirina Ramaharo_, Aug 11 2018
%D Louis H. Kauffman, Knots and Physics (Third Edition), World Scientific, 2001. See p. 353.
%H Amotz Bar-Noy, <a href="https://web.archive.org/web/20180417043918/http://www.sci.brooklyn.cuny.edu/~amotz/GC-ALGORITHMS/PRESENTATIONS/chromatic.pdf">Graph Algorithms, Chromatic Polynomials</a>.
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1911.04528">Note on sequences A123192, A137396 and A300453</a>, arXiv:1911.04528 [math.CO], 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChromaticPolynomial.html">Chromatic Polynomial</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CycleGraph.html">Cycle Graph</a>.
%F p(x;n) = (x - 2)*p(x;n-1) + (x - 1)*p(x;n-2).
%F From _Franck Maminirina Ramaharo_, Aug 11 2018: (Start)
%F T(n,0) = 0 for n > 0, and T(n,1) = (n-1)*(-1)^(n-1) for n > 1.
%F T(n,k) = (-1)^(n - k)*binomial(n,k) for k > 1. (End)
%e Triangle begins:
%e n\k| 0 1 2 3 4 5 6 7 8 9 10 11
%e ----------------------------------------------------------------
%e 1 | 0
%e 2 | 0 -1 1
%e 3 | 0 2 -3 1
%e 4 | 0 -3 6 -4 1
%e 5 | 0 4 -10 10 -5 1
%e 6 | 0 -5 15 -20 15 -6 1
%e 7 | 0 6 -21 35 -35 21 -7 1
%e 8 | 0 -7 28 -56 70 -56 28 -8 1
%e 9 | 0 8 -36 84 -126 126 -84 36 -9 1
%e 10 | 0 -9 45 -120 210 -252 210 -120 45 -10 1
%e 11 | 0 10 -55 165 -330 462 -462 330 -165 55 -11 1
%e ... reformatted and extended. - _Franck Maminirina Ramaharo_, Aug 11 2018
%o (Maxima)
%o t(n, k) := ratcoef((x - 1)^n + (-1)^n*(x - 1), x, k)$
%o T:[0]$
%o for n:2 thru 11 do T:append(T, makelist(t(n, k), k, 0, n))$
%o T; /* _Franck Maminirina Ramaharo_, Aug 11 2018 */
%Y Cf. A123192, A300453.
%K tabf,sign
%O 1,6
%A _Roger L. Bagula_, Apr 10 2008
%E Edited, new name, and corrected by _Franck Maminirina Ramaharo_, Aug 11 2018
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