%I #19 Feb 08 2017 19:06:41

%S 1,0,1,-5,8,-4,0,1,-16,112,-448,1120,-1791,1786,-1012,248,0,1,-33,510,

%T -4898,32703,-160859,602408,-1749715,3975561,-7068408,9755858,

%U -10265148,7968348,-4304712,1445104,-226720,0,1,-56,1508,-25992,321994,-3051871,23000726,-141421592,722137763,-3101089711

%N 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.

%C T differs from A212194 first at (n,k) = (5,10): T(5,10) = -3101089711, A212194(5,10) = -3101089710.

%C 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.

%H Alois P. Heinz, <a href="/A212162/b212162.txt">Rows n = 1..8, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>

%e 3 example graphs: o--o--o

%e . | /| /|

%e . |/ |/ |

%e . o--o o--o--o

%e . | /| | /| /|

%e . |/ | |/ |/ |

%e . o o--o o--o--o

%e Graph: RH_(1,1) RH_(2,2) RH_(3,3)

%e Vertices: 1 4 9

%e Edges: 0 5 16

%e 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].

%e Triangle T(n,k) begins:

%e 1, 0;

%e 1, -5, 8, -4, 0;

%e 1, -16, 112, -448, 1120, -1791, ...

%e 1, -33, 510, -4898, 32703, -160859, ...

%e 1, -56, 1508, -25992, 321994, -3051871, ... , -3101089711, ...

%e 1, -85, 3520, -94620, 1855860, -28306676, ...

%e 1, -120, 7068, -272344, 7720110, -171656543, ...

%e 1, -161, 12782, -667058, 25738055, -783003395, ...

%Y Columns 1-2 give: A000012, (-1)*A045944(n-1).

%Y Row sums (for n>1) and last elements of rows give: A000004, row lengths give: A002522.

%Y Cf. A000290, A212163, A212194.

%K sign,tabf

%O 1,4

%A _Alois P. Heinz_, May 02 2012