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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.
6
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
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
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
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, -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).
Sequence in context: A066891 A087231 A019211 * A182368 A185442 A204174
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