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A212084 Triangle T(n,k), n>=1, 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. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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 = 1..90, flattened

Eric Weisstein's World of Mathematics, Complete Bipartite Graph

Wikipedia, Chromatic Polynomial

FORMULA

T(n,k) = [q^(2n-k)] Sum_{j=1..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=1..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: A000004, 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

STATUS

approved

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Last modified August 10 19:52 EDT 2020. Contains 336381 sequences. (Running on oeis4.)