

A123970


Triangle read by rows: T(0,0)=1; T(n,k) is the coefficient of x^(nk) 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
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OFFSET

0,5


COMMENTS

This sequence is the same as A129818 up to sign.  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 JonesWenzl projector in the TemperleyLieb 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/(1x), x/(1x)^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 TemperleyLieb algebra, arXiv:0711.0016 [math.CO], 20072008.
Donatella Merlini and Renzo Sprugnoli, Arithmetic into geometric progressions through Riordan arrays, Discrete Mathematics 340.2 (2017): 160174. See page 161.
Eric Weisstein's World of Mathematics, Beraha Constants


FORMULA

f(n,x) = (2x1)f(n1,x)x^2*f(n2,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) = (2x1)f(n1,x)x^2*f(n2,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 TutteBeraha 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(n1,k)T(n1,k1)T(n2,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, nk): 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[%]


PROG

(MAGMA) /* As triangle */ [[(1)^k*Binomial(n + k, 2*k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jan 04 2019


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



