A137396 Triangle read by rows: row n gives the coefficients in the expansion of the chromatic polynomial of the n-cycle graphs.
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, 6, -21, 35, -35, 21, -7, 1, 0, -7, 28, -56, 70, -56, 28, -8, 1, 0, 8, -36, 84, -126, 126, -84, 36, -9, 1, 0, -9, 45, -120, 210, -252, 210, -120, 45, -10, 1, 0, 10
Offset: 1
Examples
Triangle begins: n\k| 0 1 2 3 4 5 6 7 8 9 10 11 ---------------------------------------------------------------- 1 | 0 2 | 0 -1 1 3 | 0 2 -3 1 4 | 0 -3 6 -4 1 5 | 0 4 -10 10 -5 1 6 | 0 -5 15 -20 15 -6 1 7 | 0 6 -21 35 -35 21 -7 1 8 | 0 -7 28 -56 70 -56 28 -8 1 9 | 0 8 -36 84 -126 126 -84 36 -9 1 10 | 0 -9 45 -120 210 -252 210 -120 45 -10 1 11 | 0 10 -55 165 -330 462 -462 330 -165 55 -11 1 ... reformatted and extended. - _Franck Maminirina Ramaharo_, Aug 11 2018
References
- Louis H. Kauffman, Knots and Physics (Third Edition), World Scientific, 2001. See p. 353.
Links
- Amotz Bar-Noy, Graph Algorithms, Chromatic Polynomials.
- Franck Ramaharo, Note on sequences A123192, A137396 and A300453, arXiv:1911.04528 [math.CO], 2019.
- Eric Weisstein's World of Mathematics, Chromatic Polynomial
- Eric Weisstein's World of Mathematics, Cycle Graph.
Programs
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Maxima
t(n, k) := ratcoef((x - 1)^n + (-1)^n*(x - 1), x, k)$ T:[0]$ for n:2 thru 11 do T:append(T, makelist(t(n, k), k, 0, n))$ T; /* Franck Maminirina Ramaharo, Aug 11 2018 */
Formula
p(x;n) = (x - 2)*p(x;n-1) + (x - 1)*p(x;n-2).
From Franck Maminirina Ramaharo, Aug 11 2018: (Start)
T(n,0) = 0 for n > 0, and T(n,1) = (n-1)*(-1)^(n-1) for n > 1.
T(n,k) = (-1)^(n - k)*binomial(n,k) for k > 1. (End)
Extensions
Edited, new name, and corrected by Franck Maminirina Ramaharo, Aug 11 2018
Comments