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A123970 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). 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

This sequence is the same as A129818. - T. D. Noe, Sep 30 2011

Fendley and Krushkal: "One of the remarkable features of the chromatic polynomial chi(Q) is Tutte's golden identity. This relates chi(phi+2) for any triangulation of the sphere to (chi(phi+1))^2 for the same graph, where phi denotes the golden ratio. We show that this result fits in the framework of quantum topology and give a proof of Tutte's identity using the notion of the chromatic algebra, whose Markov trace is the chromatic polynomial of an associated graph. We also show that another relation of Tutte's for the chromatic polynomial at Q=phi+1 precisely corresponds to a Jones-Wenzl projector in the Temperley-Lieb algebra. We show that such a relation exists whenever Q = 2+2cos(2 pi j/(n+1)) for j<n positive integers. When j=1, these are the Beraha numbers and in this case the existence of such a relation was conjectured by Tutte. We present a recursive formula for this sequence of chromatic polynomial relations. - Jonathan Vos Post, Nov 04 2007

Riordan array (1/(1-x), -x/(1-x)^2). - Philippe Deléham, Feb 18 2012

REFERENCES

Beraha, S., Ph.D. thesis. Baltimore, MD: Johns Hopkins University, 1974.

S. R. Finch, Mathematical Constants (Encyclopedia of Mathematics and its Applications), chapter 5.25.

Tutte, W. T., "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.

LINKS

Table of n, a(n) for n=0..71.

Paul Fendley and Vyacheslav Krushkal, Tutte chromatic identities and the Temperley-Lieb algebra, arXiv:0711.0016 [math.CO], 2007-2008.

Donatella Merlini and Renzo Sprugnoli, Arithmetic into geometric progressions through Riordan arrays, Discrete Mathematics 340.2 (2017): 160-174. See page 161.

Eric Weisstein's World of Mathematics, Beraha Constants

FORMULA

f(n,x) = (2x-1)f(n-1,x)-x^2*f(n-2,x), where f(n,x) is the monic characteristic polynomial of the n X n matrix from the definition and f(0,x)=1.

f(n,x) = (2x-1)f(n-1,x)-x^2*f(n-2,x), where f(n,x) is the monic characteristic polynomial of the n X n matrix from the definition and f(0,x)=1. See formula in Fendley and Krushkal, generalizing the Tutte-Beraha constants. - Jonathan Vos Post, Nov 04 2007

T(n,k) = (-1)^k * A085478(n,k) = (-1)^n * A129818(n,k). - Philippe Deléham, Feb 06 2012

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

EXAMPLE

Triangular sequence (gives the odd Tutte -Beraha constants as roots!)

{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}

MAPLE

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

MATHEMATICA

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[%]

CROSSREFS

Cf. A085478, A076756.

Cf. A109954, A129818, A143858, A165253. - R. J. Mathar, Jan 10 2011

Modulo signs, inverse matrix to A039599.

Sequence in context: A103141 A129818 A085478 * A055898 A145904 A273350

Adjacent sequences:  A123967 A123968 A123969 * A123971 A123972 A123973

KEYWORD

sign,tabl

AUTHOR

Gary W. Adamson and Roger L. Bagula, Oct 29 2006

EXTENSIONS

Edited by N. J. A. Sloane, Nov 29 2006

STATUS

approved

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Last modified February 20 04:19 EST 2018. Contains 299358 sequences. (Running on oeis4.)