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A123970
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Triangle read by rows: T(0,0)=1; T(n,k) is the coefficient of x^(n-k) in the monic characteristic polynomial of the n X n matrix (min(i,j)) (i,j=1,2,...,n) (0 <= k <= n, n >= 1).
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3
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1, 1, -1, 1, -3, 1, 1, -6, 5, -1, 1, -10, 15, -7, 1, 1, -15, 35, -28, 9, -1, 1, -21, 70, -84, 45, -11, 1, 1, -28, 126, -210, 165, -66, 13, -1, 1, -36, 210, -462, 495, -286, 91, -15, 1, 1, -45, 330, -924, 1287, -1001, 455, -120, 17, -1, 1, -55, 495, -1716, 3003, -3003, 1820, -680, 153, -19, 1, 1, -66, 715, -3003, 6435, -8008
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OFFSET
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0,5
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COMMENTS
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REFERENCES
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S. Beraha, Infinite non-trivial families of maps and chromials, Ph.D. thesis. Baltimore, MD: Johns Hopkins University, 1975.
S. R. Finch, Mathematical Constants (Encyclopedia of Mathematics and its Applications), chapter 5.25.
W. T. Tutte, "More about Chromatic Polynomials and the Golden Ratio." In Combinatorial Structures and their Applications: Proc. Calgary Internat. Conf., Calgary, Alberta, 1969. New York: Gordon and Breach, p. 439, 1969.
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LINKS
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FORMULA
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f(n,x) = (2x-1)f(n-1,x)-x^2*f(n-2,x), where f(n,x) is the characteristic polynomial of the n X n matrix from the definition and f(0,x)=1. See formula in Fendley and Krushkal. - Jonathan Vos Post, Nov 04 2007
T(n,k) = 2*T(n-1,k) - T(n-1,k-1) - T(n-2,k), T(0,0)=T(1,0)=1, T(1,1)=-1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Nov 29 2013
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EXAMPLE
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Triangular sequence (gives the odd Tutte-Beraha constants as roots!) begins:
1;
1, -1;
1, -3, 1;
1, -6, 5, -1;
1, -10, 15, -7, 1;
1, -15, 35, -28, 9, -1;
1, -21, 70, -84, 45, -11, 1;
1, -28, 126, -210, 165, -66, 13, -1;
1, -36, 210, -462, 495, -286, 91, -15, 1;
1, -45, 330, -924, 1287, -1001, 455, -120, 17, -1;
...
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MAPLE
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with(linalg): m:=(i, j)->min(i, j): M:=n->matrix(n, n, m): T:=(n, k)->coeff(charpoly(M(n), x), x, n-k): 1; for n from 1 to 11 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
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MATHEMATICA
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An[d_] := MatrixPower[Table[Min[n, m], {n, 1, d}, {m, 1, d}], -1]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d, 1, 20}]]; Flatten[%]
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PROG
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(Magma) /* As triangle */ [[(-1)^k*Binomial(n + k, 2*k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jan 04 2019
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CROSSREFS
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Modulo signs, inverse matrix to A039599.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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