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A266972 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.

%I A266972 #24 Feb 16 2025 08:33:29
%S A266972 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 A266972 0,1,-12,66,-202,351,-319,115,0,1,-16,120,-524,1400,-2236,1930,-675,0,
%U A266972 1,-20,190,-1080,3925,-9164,13186,-10489,3451,0,1,-25,300,-2200,10650,-34730,75170,-102545,78610,-25231,0
%N A266972 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 A266972 The (n,2)-Turán graph is also the complete bipartite graph K_{floor(n/2),ceiling(n/2)}.
%H A266972 Alois P. Heinz, <a href="/A266972/b266972.txt">Rows n = 0..140, flattened</a>
%H A266972 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%H A266972 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>
%H A266972 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tur%C3%A1n_graph">Turán graph</a>
%F A266972 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 A266972 Sum_{k=0..n} abs(T(n,k)) = A266695(n).
%e A266972 Triangle T(n,k) begins:
%e A266972   1;
%e A266972   1,   0;
%e A266972   1,  -1,   0;
%e A266972   1,  -2,   1,    0;
%e A266972   1,  -4,   6,   -3,    0;
%e A266972   1,  -6,  15,  -17,    7,     0;
%e A266972   1,  -9,  36,  -75,   78,   -31,    0;
%e A266972   1, -12,  66, -202,  351,  -319,  115,    0;
%e A266972   1, -16, 120, -524, 1400, -2236, 1930, -675,  0;
%e A266972   ...
%p A266972 P:= n-> (h-> expand(add(Stirling2(h, j)*mul(q-i,
%p A266972     i=0..j-1)*(q-j)^(n-h), j=0..h)))(iquo(n, 2)):
%p A266972 T:= n-> (p-> seq(coeff(p, q, n-i), i=0..n))(P(n)):
%p A266972 seq(T(n), n=0..12);
%Y A266972 Columns k=0-1 give: A000012, (-1)*A002620.
%Y A266972 Main diagonal gives A000007.
%Y A266972 Cf. A212084, A266695.
%K A266972 sign,tabl
%O A266972 0,8
%A A266972 _Alois P. Heinz_, Jan 07 2016