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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 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 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 jn. - 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[%] 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 A085478 A129818 * 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 October 16 13:51 EDT 2019. Contains 328093 sequences. (Running on oeis4.)