OFFSET
1,6
COMMENTS
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
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.
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)
EXAMPLE
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
PROG
(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 */
CROSSREFS
KEYWORD
tabf,sign
AUTHOR
Roger L. Bagula, Apr 10 2008
EXTENSIONS
Edited, new name, and corrected by Franck Maminirina Ramaharo, Aug 11 2018
STATUS
approved