%I #23 May 03 2024 09:56:34

%S 1,1,0,1,-1,0,1,-2,1,0,1,-4,6,-3,0,1,-6,15,-17,7,0,1,-9,36,-75,78,-31,

%T 0,1,-12,66,-202,351,-319,115,0,1,-16,120,-524,1400,-2236,1930,-675,0,

%U 1,-20,190,-1080,3925,-9164,13186,-10489,3451,0,1,-25,300,-2200,10650,-34730,75170,-102545,78610,-25231,0

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

%C The (n,2)-Turán graph is also the complete bipartite graph K_{floor(n/2),ceiling(n/2)}.

%H Alois P. Heinz, <a href="/A266972/b266972.txt">Rows n = 0..140, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tur%C3%A1n_graph">Turán graph</a>

%F T(n,k) = [q^(n-k)] Sum_{j=0..floor(n/2)} (q-j)^(n-floor(n/2)) * Stirling2(floor(n/2),j) * Product_{i=0..j-1} (q-i).

%F Sum_{k=0..n} abs(T(n,k)) = A266695(n).

%e Triangle T(n,k) begins:

%e 1;

%e 1, 0;

%e 1, -1, 0;

%e 1, -2, 1, 0;

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

%e 1, -6, 15, -17, 7, 0;

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

%e 1, -12, 66, -202, 351, -319, 115, 0;

%e 1, -16, 120, -524, 1400, -2236, 1930, -675, 0;

%e ...

%p P:= n-> (h-> expand(add(Stirling2(h, j)*mul(q-i,

%p i=0..j-1)*(q-j)^(n-h), j=0..h)))(iquo(n, 2)):

%p T:= n-> (p-> seq(coeff(p, q, n-i), i=0..n))(P(n)):

%p seq(T(n), n=0..12);

%Y Columns k=0-1 give: A000012, (-1)*A002620.

%Y Main diagonal gives A000007.

%Y Cf. A212084, A266695.

%K sign,tabl

%O 0,8

%A _Alois P. Heinz_, Jan 07 2016