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A212162
Triangle T(n,k), n>=1, 0<=k<=n^2, read by rows: row n gives the coefficients of the chromatic polynomial of the rhombic hexagonal square grid graph RH_(n,n), highest powers first.
21
1, 0, 1, -5, 8, -4, 0, 1, -16, 112, -448, 1120, -1791, 1786, -1012, 248, 0, 1, -33, 510, -4898, 32703, -160859, 602408, -1749715, 3975561, -7068408, 9755858, -10265148, 7968348, -4304712, 1445104, -226720, 0, 1, -56, 1508, -25992, 321994, -3051871, 23000726, -141421592, 722137763, -3101089711
OFFSET
1,4
COMMENTS
T differs from A212194 first at (n,k) = (5,10): T(5,10) = -3101089711, A212194(5,10) = -3101089710.
The rhombic hexagonal square grid graph RH_(n,n) has n^2 = A000290(n) vertices and (n-1)*(3*n-1) = A045944(n-1) edges. The chromatic polynomial of RH_(n,n) has n^2+1 = A002522(n) coefficients.
LINKS
Alois P. Heinz, Rows n = 1..8, flattened
EXAMPLE
3 example graphs: o--o--o
. | /| /|
. |/ |/ |
. o--o o--o--o
. | /| | /| /|
. |/ | |/ |/ |
. o o--o o--o--o
Graph: RH_(1,1) RH_(2,2) RH_(3,3)
Vertices: 1 4 9
Edges: 0 5 16
The rhombic hexagonal square grid graph RH_(2,2) has chromatic polynomial q*(q-1)*(q-2)^2 = q^4 -5*q^3 +8*q^2 -4*q => row 2 = [1, -5, 8, -4, 0].
Triangle T(n,k) begins:
1, 0;
1, -5, 8, -4, 0;
1, -16, 112, -448, 1120, -1791, ...
1, -33, 510, -4898, 32703, -160859, ...
1, -56, 1508, -25992, 321994, -3051871, ... , -3101089711, ...
1, -85, 3520, -94620, 1855860, -28306676, ...
1, -120, 7068, -272344, 7720110, -171656543, ...
1, -161, 12782, -667058, 25738055, -783003395, ...
CROSSREFS
Columns 1-2 give: A000012, (-1)*A045944(n-1).
Row sums (for n>1) and last elements of rows give: A000004, row lengths give: A002522.
Sequence in context: A019907 A248007 A212194 * A249445 A199450 A366072
KEYWORD
sign,tabf
AUTHOR
Alois P. Heinz, May 02 2012
STATUS
approved