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%I #53 Feb 08 2022 08:10:41
%S 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,
%T 45,-11,1,1,-28,126,-210,165,-66,13,-1,1,-36,210,-462,495,-286,91,-15,
%U 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
%N 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).
%C This sequence is the same as A129818 up to sign. - _T. D. Noe_, Sep 30 2011
%C Riordan array (1/(1-x), -x/(1-x)^2). - _Philippe Deléham_, Feb 18 2012
%D S. Beraha, Infinite non-trivial families of maps and chromials, Ph.D. thesis. Baltimore, MD: Johns Hopkins University, 1975.
%D S. R. Finch, Mathematical Constants (Encyclopedia of Mathematics and its Applications), chapter 5.25.
%D 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.
%H Paul Fendley and Vyacheslav Krushkal, <a href="http://arXiv.org/abs/0711.0016">Tutte chromatic identities and the Temperley-Lieb algebra</a>, arXiv:0711.0016 [math.CO], 2007-2008.
%H Donatella Merlini and Renzo Sprugnoli, <a href="https://doi.org/10.1016/j.disc.2016.08.017">Arithmetic into geometric progressions through Riordan arrays</a>, Discrete Mathematics 340.2 (2017): 160-174. See page 161.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BerahaConstants.html">Beraha Constants</a>
%F 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
%F T(n,k) = (-1)^k * A085478(n,k) = (-1)^n * A129818(n,k). - _Philippe Deléham_, Feb 06 2012
%F 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
%e Triangular sequence (gives the odd Tutte-Beraha constants as roots!) begins:
%e 1;
%e 1, -1;
%e 1, -3, 1;
%e 1, -6, 5, -1;
%e 1, -10, 15, -7, 1;
%e 1, -15, 35, -28, 9, -1;
%e 1, -21, 70, -84, 45, -11, 1;
%e 1, -28, 126, -210, 165, -66, 13, -1;
%e 1, -36, 210, -462, 495, -286, 91, -15, 1;
%e 1, -45, 330, -924, 1287, -1001, 455, -120, 17, -1;
%e ...
%p 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
%t 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[%]
%o (Magma) /* As triangle */ [[(-1)^k*Binomial(n + k, 2*k): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Jan 04 2019
%Y Cf. A085478, A076756.
%Y Cf. A109954, A129818, A143858, A165253. - _R. J. Mathar_, Jan 10 2011
%Y Modulo signs, inverse matrix to A039599.
%K sign,tabl
%O 0,5
%A _Gary W. Adamson_ and _Roger L. Bagula_, Oct 29 2006
%E Edited by _N. J. A. Sloane_, Nov 29 2006
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