A212194 Triangle T(n,k), n>=1, 0<=k<=n^2, read by rows: row n gives the coefficients of the chromatic polynomial of the staggered hexagonal square grid graph SH_(n,n), highest powers first.

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, -3101089710

Offset: 1

Author

Alois P. Heinz, May 03 2012

Keywords

• sign
• tabf

Comments

T differs from A212162 first at (n,k) = (5,10): T(5,10) = -3101089710, A212162(5,10) = -3101089711.

The staggered hexagonal square grid graph SH_(n,n) has n^2 = A000290(n) vertices and (n-1)*(3*n-1) = A045944(n-1) edges. The chromatic polynomial of SH_(n,n) has n^2+1 = A002522(n) coefficients.

Examples

			3 example graphs:                        o--o--o
.                                        | /|\ |
.                                        |/ | \|
.                            o--o        o--o--o
.                            | /|        | /|\ |
.                            |/ |        |/ | \|
.               o            o--o        o--o--o
Graph:       SH_(1,1)      SH_(2,2)      SH_(3,3)
Vertices:       1             4             9
Edges:          0             5            16
The staggered hexagonal square grid graph SH_(2,2) has chromatic polynomial 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, ... , -3101089710, ...
1,  -85,  3520,  -94620,  1855860,  -28306676, ...
1, -120,  7068, -272344,  7720110, -171656543, ...
1, -161, 12782, -667058, 25738055, -783003395, ...
		

Links

• Alois P. Heinz, Rows n = 1..8, flattened
• Wikipedia, Chromatic Polynomial

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.

Cf. A000290, A212162, A212195.