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Triangle read by rows, characteristic polynomials of Cartan ring matrices.
1

%I #36 Aug 08 2018 15:12:27

%S 1,1,-2,1,-4,3,1,-6,9,-4,1,-8,20,-16,4,1,-10,35,-50,25,-4,1,-12,54,

%T -112,105,-36,4,1,-14,77,-210,294,-196,49,-4,1,-16,104,-352,660,-672,

%U 336,-64,4,1,-18,135,-546,1287,-1782,1386,-540,81,-4,1,-20,170,-800,2275,-4004,4290,-2640,825,-100,4

%N Triangle read by rows, characteristic polynomials of Cartan ring matrices.

%D William G. Harter, University of Arkansas; personal communication

%H P. Damianou , <a href="http://arxiv.org/abs/1110.6620">On the characteristic polynomials of Cartan matrices and Chebyshev polynomials</a>, arXiv preprint arXiv:1110.6620 [math.RT], 2014.

%H P. Damianou and C. Evripidou, <a href="http://arxiv.org/abs/1409.3956">Characteristic and Coxeter polynomials for affine Lie algebras</a>, arXiv preprint arXiv:1409.3956 [math.RT], 2014.

%H Todd Rowland, Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CartanMatrix.html">Cartan Matrix</a>

%F Triangle read by rows, n-th row = characteristic polynomial of n X n Cartan ring matrix, defined as a Cartan matrix with 1's in the upper right and lower left corners, i.e., positions (1,n) and (n,1).

%F The coefficients of characteristic polynomials of matrices C_n, defined by

%F C_n=

%F (2 -1 0 ... 0 1)

%F (-1 2 -1 0 ... 0)

%F (0 -1 2 -1 0 ... 0)

%F ...

%F (0 ... 0 -1 2 -1)

%F (1 0 ... 0 -1 2),

%F give the same triangle T(n,k), for n>0, k=0,...,n, with T(0,0)=1. - _L. Edson Jeffery_, Mar 27 2011

%F It appears that for n >= 3 the n-th row polynomial equals 2*T(2*n,sqrt(x)/2) + 2*(-1)^n, where T(n,x) denotes the Chebyshev polynomial of the first kind (A008310). Checked for n = 3 through n = 12. - _Peter Bala_, May 04 2014

%F Apparently, omitting the diagonal here, this triangular array is signed, reversed A156308 (cf. A127677, A217476, A263916). For relations among the characteristic polynomials of Cartan matrices of the Coxeter root groups, Chebyshev polynomials, cyclotomic polynomials, and the polynomials of this entry, see Damianou (p. 12, 20, and 21) and Damianou and Evripidou (p. 7). - _Tom Copeland_, Nov 07 2015

%e Triangle begins:

%e 1;

%e 1, -2;

%e 1, -4, 3;

%e 1, -6, 9, -4;

%e 1, -8, 20, -16, 4;

%e 1, -10, 35, -50, 25, -4;

%e 1, -12, 54, -112, 105, -36, 4;

%e 1, -14, 77, -210, 294, -196, 49, -4;

%e 1, -16, 104, -352, 660, -672, 336, -64, 4;

%e 1, -18, 135, -546, 1287, -1782, 1386, -540, 81, -4;

%e 1, -20, 170, -800, 2275, -4004, 4290, -2640, 825, -100, 4;

%e ...

%e Example: x^5 -10x^4 + 35x^3 -50x^2 + 25x - 4 = (x - 4) * (x^2 - 3x + 1)^2 is the characteristic polynomial of the matrix

%e [ 2,-1, 0, 0, 1]

%e [-1, 2,-1, 0, 0]

%e [ 0,-1, 2,-1, 0]

%e [ 0, 0,-1, 2,-1]

%e [ 1, 0, 0,-1, 2].

%t M[n_] := SparseArray[{Band[{1, 1}] -> 2, Band[{1, 2}] -> -1, Band[{2, 1}] -> -1, {1, n} -> 1, {n, 1} -> 1}, {n, n}];

%t row[0] = {1}; row[1] = {1, -2};

%t row[n_] := (-1)^n CharacteristicPolynomial[M[n], x] // CoefficientList[#, x]& // Reverse;

%t Table[row[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Aug 08 2018 *)

%Y Cf. A008310.

%Y Cf. A156308, A127677, A217476, A263916.

%K tabl,sign

%O 0,3

%A _Gary W. Adamson_ & _Roger L. Bagula_, Nov 22 2008

%E Edited by _L. Edson Jeffery_, Mar 26 2011

%E Some terms corrected from _Peter Bala_, May 04 2014