A212084 - OEIS

A212084

Triangle T(n,k), n>=0, 0<=k<=2n, read by rows: row n gives the coefficients of the chromatic polynomial of the complete bipartite graph K_(n,n), highest powers first.

1, 1, -1, 0, 1, -4, 6, -3, 0, 1, -9, 36, -75, 78, -31, 0, 1, -16, 120, -524, 1400, -2236, 1930, -675, 0, 1, -25, 300, -2200, 10650, -34730, 75170, -102545, 78610, -25231, 0, 1, -36, 630, -6915, 52080, -279142, 1074822, -2942445, 5552680, -6796926, 4787174

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OFFSET

0,6

COMMENTS

The complete bipartite graph K_(n,n) has 2n vertices and n^2 = A000290(n) edges. The chromatic polynomial of K_(n,n) has 2n+1 = A005408(n) coefficients.

LINKS

Alois P. Heinz, Rows n = 0..90, flattened

Eric Weisstein's World of Mathematics, Complete Bipartite Graph

Wikipedia, Chromatic Polynomial

FORMULA

T(n,k) = [q^(2n-k)] Sum_{j=0..n} (q-j)^n * S2(n,j) * Product_{i=0..j-1} (q-i).

EXAMPLE

3 example graphs: +-----------+

. o o o o o o |

. | |\ /| |\ /|\ /|\ /

. | | X | | X | X | X

. | |/ \| |/ \|/ \|/ \

. o o o o o o |

. +-----------+

Graph: K_(1,1) K_(2,2) K_(3,3)

Vertices: 2 4 6

Edges: 1 4 9

The complete bipartite graph K_(2,2) is the cycle graph C_4 with chromatic polynomial q^4 -4*q^3 +6*q^2 -3*q => row 2 = [1, -4, 6, -3, 0].

Triangle T(n,k) begins:

1, -1, 0;

1, -4, 6, -3, 0;

1, -9, 36, -75, 78, -31, 0;

1, -16, 120, -524, 1400, -2236, 1930, -675, ...

1, -25, 300, -2200, 10650, -34730, 75170, -102545, ...

1, -36, 630, -6915, 52080, -279142, 1074822, -2942445, ...

...

MAPLE

P:= n-> add(Stirling2(n, k) *mul(q-i, i=0..k-1) *(q-k)^n, k=0..n):

T:= n-> seq(coeff(P(n), q, 2*n-k), k=0..2*n):

seq(T(n), n=1..8);

CROSSREFS

Columns k=0-2 give: A000012, (-1)*A000290, A083374.

Row sums and last elements of rows give: A000007.

Row lengths give: A005408.

Sums of absolute values of row elements give: A048163(n+1).

T(n,2n-1) = (-1)*A092552(n).

Cf. A008277, A212085, A182368, A185442, A193233, A193277, A193283, A266695, A266972.

Sequence in context: A066891 A087231 A019211 * A182368 A185442 A204174

Adjacent sequences: A212081 A212082 A212083 * A212085 A212086 A212087

KEYWORD

sign,tabf

AUTHOR

Alois P. Heinz, Apr 30 2012

EXTENSIONS

T(0,0)=1 prepended by Alois P. Heinz, May 03 2024

STATUS

approved