This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A137396 #37 Feb 16 2025 08:33:07 %S A137396 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 A137396 6,-21,35,-35,21,-7,1,0,-7,28,-56,70,-56,28,-8,1,0,8,-36,84,-126,126, %U A137396 -84,36,-9,1,0,-9,45,-120,210,-252,210,-120,45,-10,1,0,10 %N A137396 Triangle read by rows: row n gives the coefficients in the expansion of the chromatic polynomial of the n-cycle graphs. %C A137396 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 A137396 Louis H. Kauffman, Knots and Physics (Third Edition), World Scientific, 2001. See p. 353. %H A137396 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 A137396 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 A137396 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChromaticPolynomial.html">Chromatic Polynomial</a> %H A137396 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CycleGraph.html">Cycle Graph</a>. %F A137396 p(x;n) = (x - 2)*p(x;n-1) + (x - 1)*p(x;n-2). %F A137396 From _Franck Maminirina Ramaharo_, Aug 11 2018: (Start) %F A137396 T(n,0) = 0 for n > 0, and T(n,1) = (n-1)*(-1)^(n-1) for n > 1. %F A137396 T(n,k) = (-1)^(n - k)*binomial(n,k) for k > 1. (End) %e A137396 Triangle begins: %e A137396 n\k| 0 1 2 3 4 5 6 7 8 9 10 11 %e A137396 ---------------------------------------------------------------- %e A137396 1 | 0 %e A137396 2 | 0 -1 1 %e A137396 3 | 0 2 -3 1 %e A137396 4 | 0 -3 6 -4 1 %e A137396 5 | 0 4 -10 10 -5 1 %e A137396 6 | 0 -5 15 -20 15 -6 1 %e A137396 7 | 0 6 -21 35 -35 21 -7 1 %e A137396 8 | 0 -7 28 -56 70 -56 28 -8 1 %e A137396 9 | 0 8 -36 84 -126 126 -84 36 -9 1 %e A137396 10 | 0 -9 45 -120 210 -252 210 -120 45 -10 1 %e A137396 11 | 0 10 -55 165 -330 462 -462 330 -165 55 -11 1 %e A137396 ... reformatted and extended. - _Franck Maminirina Ramaharo_, Aug 11 2018 %o A137396 (Maxima) %o A137396 t(n, k) := ratcoef((x - 1)^n + (-1)^n*(x - 1), x, k)$ %o A137396 T:[0]$ %o A137396 for n:2 thru 11 do T:append(T, makelist(t(n, k), k, 0, n))$ %o A137396 T; /* _Franck Maminirina Ramaharo_, Aug 11 2018 */ %Y A137396 Cf. A123192, A300453. %K A137396 tabf,sign %O A137396 1,6 %A A137396 _Roger L. Bagula_, Apr 10 2008 %E A137396 Edited, new name, and corrected by _Franck Maminirina Ramaharo_, Aug 11 2018