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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

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

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

Eric Weisstein's World of Mathematics, Complete Bipartite Graph

Wikipedia, Chromatic Polynomial

Wikipedia, Turán graph

FORMULA

T(n,k) = [q^(n-k)] Sum_{j=1..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).

EXAMPLE

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;

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);

CROSSREFS

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

Main diagonal gives A000007.

Cf. A212084, A266695.

Sequence in context: A323174 A295683 A165519 * A266493 A075374 A293024

Adjacent sequences:  A266969 A266970 A266971 * A266973 A266974 A266975

KEYWORD

sign,tabl

AUTHOR

Alois P. Heinz, Jan 07 2016

STATUS

approved

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Last modified July 11 08:10 EDT 2020. Contains 335626 sequences. (Running on oeis4.)