A212208 Triangle T(n,k), n>=1, 0<=k<=n^2, read by rows: row n gives the coefficients of the chromatic polynomial of the square diagonal grid graph DG_(n,n), highest powers first.

1, 0, 1, -6, 11, -6, 0, 1, -20, 174, -859, 2627, -5082, 6048, -4023, 1134, 0, 1, -42, 825, -10054, 85011, -528254, 2491825, -9084089, 25795983, -57031153, 97292827, -125639547, 118705077, -77301243, 30931875, -5709042, 0, 1, -72, 2492, -55183, 877812

Offset

1,4

Comments

The square diagonal grid graph DG_(n,n) has n^2 = A000290(n) vertices and 2*(n-1)*(2*n-1) = A002943(n-1) edges. The chromatic polynomial of DG_(n,n) has n^2+1 = A002522(n) coefficients.

Links

Alois P. Heinz, Rows n = 1..7, flattened

Wikipedia, Chromatic Polynomial

Example

3 example graphs:                          o---o---o
.                                          |\ /|\ /|
.                                          | X | X |
.                                          |/ \|/ \|
.                             o---o        o---o---o
.                             |\ /|        |\ /|\ /|
.                             | X |        | X | X |
.                             |/ \|        |/ \|/ \|
.                o            o---o        o---o---o
Graph:        DG_(1,1)       DG_(2,2)       DG_(3,3)
Vertices:        1              4              9
Edges:           0              6             20
The square diagonal grid graph DG_(2,2) equals the complete graph K_4 and has chromatic polynomial q*(q-1)*(q-2)*(q-3) = q^4 -6*q^3 +11*q^2 -6*q => row 2 = [1, -6, 11, -6, 0].
Triangle T(n,k) begins:
1,    0;
1,   -6,    11,      -6,        0;
1,  -20,   174,    -859,     2627,      -5082, ...
1,  -42,   825,  -10054,    85011,    -528254, ...
1,  -72,  2492,  -55183,   877812,  -10676360, ...
1, -110,  5895, -205054,  5203946, -102687204, ...
1, -156, 11946, -598491, 22059705, -637802510, ...

Crossrefs

Columns 1-2 give: A000012, (-1)*A002943(n-1).

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

Cf. A000290, A212209.

Sequence in context: A009443 A258054 A106540 * A334280 A134012 A103704

Adjacent sequences: A212205 A212206 A212207 * A212209 A212210 A212211

Keyword

sign,tabf

Author

Alois P. Heinz, May 04 2012

Status

approved