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A266972
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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.
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4
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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
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OFFSET
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0,8
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COMMENTS
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The (n,2)-Turán graph is also the complete bipartite graph K_{floor(n/2),ceiling(n/2)}.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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;
...
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MAPLE
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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);
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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