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.

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, 0, 1, -12, 66, -202, 351, -319, 115, 0, 1, -16, 120, -524, 1400, -2236, 1930, -675, 0, 1, -20, 190, -1080, 3925, -9164, 13186, -10489, 3451, 0, 1, -25, 300, -2200, 10650, -34730, 75170, -102545, 78610, -25231, 0

Offset: 0

Author

Alois P. Heinz, Jan 07 2016

Keywords

• sign
• tabl

Comments

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

Examples

			Triangle T(n,k) begins:
  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,    0;
  1, -12,  66, -202,  351,  -319,  115,    0;
  1, -16, 120, -524, 1400, -2236, 1930, -675,  0;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Eric Weisstein's World of Mathematics, Complete Bipartite Graph
• Wikipedia, Chromatic Polynomial
• Wikipedia, Turán graph

Crossrefs

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

Main diagonal gives A000007.

Cf. A212084, A266695.

Programs

Maple

P:= n-> (h-> expand(add(Stirling2(h, j)*mul(q-i,
    i=0..j-1)*(q-j)^(n-h), j=0..h)))(iquo(n, 2)):
T:= n-> (p-> seq(coeff(p, q, n-i), i=0..n))(P(n)):
seq(T(n), n=0..12);

Formula

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).

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