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

1,6

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

Table of n, a(n) for n=1..66.

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

Cf. A123192, A300453.

Sequence in context: A352799 A082839 A130717 * A244213 A346415 A362894

Adjacent sequences: A137393 A137394 A137395 * A137397 A137398 A137399

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