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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.

%I A212084 #38 Feb 16 2025 08:33:17
%S A212084 1,1,-1,0,1,-4,6,-3,0,1,-9,36,-75,78,-31,0,1,-16,120,-524,1400,-2236,
%T A212084 1930,-675,0,1,-25,300,-2200,10650,-34730,75170,-102545,78610,-25231,
%U A212084 0,1,-36,630,-6915,52080,-279142,1074822,-2942445,5552680,-6796926,4787174
%N 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.
%C A212084 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.
%H A212084 Alois P. Heinz, <a href="/A212084/b212084.txt">Rows n = 0..90, flattened</a>
%H A212084 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%H A212084 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>
%F A212084 T(n,k) = [q^(2n-k)] Sum_{j=0..n} (q-j)^n * S2(n,j) * Product_{i=0..j-1} (q-i).
%e A212084 3 example graphs:                     +-----------+
%e A212084 .                 o        o   o      o   o   o   |
%e A212084 .                 |        |\ /|      |\ /|\ /|\ /
%e A212084 .                 |        | X |      | X | X | X
%e A212084 .                 |        |/ \|      |/ \|/ \|/ \
%e A212084 .                 o        o   o      o   o   o   |
%e A212084 .                                     +-----------+
%e A212084 Graph:         K_(1,1)    K_(2,2)      K_(3,3)
%e A212084 Vertices:         2          4            6
%e A212084 Edges:            1          4            9
%e A212084 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].
%e A212084 Triangle T(n,k) begins:
%e A212084   1;
%e A212084   1,  -1,   0;
%e A212084   1,  -4,   6,    -3,     0;
%e A212084   1,  -9,  36,   -75,    78,     -31,       0;
%e A212084   1, -16, 120,  -524,  1400,   -2236,    1930,     -675, ...
%e A212084   1, -25, 300, -2200, 10650,  -34730,   75170,  -102545, ...
%e A212084   1, -36, 630, -6915, 52080, -279142, 1074822, -2942445, ...
%e A212084   ...
%p A212084 P:= n-> add(Stirling2(n, k) *mul(q-i, i=0..k-1) *(q-k)^n, k=0..n):
%p A212084 T:= n-> seq(coeff(P(n), q, 2*n-k), k=0..2*n):
%p A212084 seq(T(n), n=1..8);
%Y A212084 Columns k=0-2 give: A000012, (-1)*A000290, A083374.
%Y A212084 Row sums and last elements of rows give: A000007.
%Y A212084 Row lengths give: A005408.
%Y A212084 Sums of absolute values of row elements give: A048163(n+1).
%Y A212084 T(n,2n-1) = (-1)*A092552(n).
%Y A212084 Cf. A008277, A212085, A182368, A185442, A193233, A193277, A193283, A266695, A266972.
%K A212084 sign,tabf
%O A212084 0,6
%A A212084 _Alois P. Heinz_, Apr 30 2012
%E A212084 T(0,0)=1 prepended by _Alois P. Heinz_, May 03 2024