cp's OEIS Frontend

A152060 Triangle read by rows, characteristic polynomials of Cartan ring matrices.

%I A152060 #37 Feb 16 2025 08:33:09
%S A152060 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 A152060 -112,105,-36,4,1,-14,77,-210,294,-196,49,-4,1,-16,104,-352,660,-672,
%U A152060 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 A152060 Triangle read by rows, characteristic polynomials of Cartan ring matrices.
%D A152060 William G. Harter, University of Arkansas; personal communication
%H A152060 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 A152060 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 A152060 Todd Rowland, Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CartanMatrix.html">Cartan Matrix</a>
%F A152060 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 A152060 The coefficients of characteristic polynomials of matrices C_n, defined by
%F A152060 C_n=
%F A152060 (2 -1 0 ... 0 1)
%F A152060 (-1 2 -1 0 ... 0)
%F A152060 (0 -1 2 -1 0 ... 0)
%F A152060 ...
%F A152060 (0 ... 0 -1 2 -1)
%F A152060 (1 0 ... 0 -1 2),
%F A152060 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 A152060 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 A152060 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 A152060 Triangle begins:
%e A152060 1;
%e A152060 1, -2;
%e A152060 1, -4, 3;
%e A152060 1, -6, 9, -4;
%e A152060 1, -8, 20, -16, 4;
%e A152060 1, -10, 35, -50, 25, -4;
%e A152060 1, -12, 54, -112, 105, -36, 4;
%e A152060 1, -14, 77, -210, 294, -196, 49, -4;
%e A152060 1, -16, 104, -352, 660, -672, 336, -64, 4;
%e A152060 1, -18, 135, -546, 1287, -1782, 1386, -540, 81, -4;
%e A152060 1, -20, 170, -800, 2275, -4004, 4290, -2640, 825, -100, 4;
%e A152060 ...
%e A152060 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 A152060 [ 2,-1, 0, 0, 1]
%e A152060 [-1, 2,-1, 0, 0]
%e A152060 [ 0,-1, 2,-1, 0]
%e A152060 [ 0, 0,-1, 2,-1]
%e A152060 [ 1, 0, 0,-1, 2].
%t A152060 M[n_] := SparseArray[{Band[{1, 1}] -> 2, Band[{1, 2}] -> -1, Band[{2, 1}] -> -1, {1, n} -> 1, {n, 1} -> 1}, {n, n}];
%t A152060 row[0] = {1}; row[1] = {1, -2};
%t A152060 row[n_] := (-1)^n CharacteristicPolynomial[M[n], x] // CoefficientList[#, x]& // Reverse;
%t A152060 Table[row[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Aug 08 2018 *)
%Y A152060 Cf. A008310.
%Y A152060 Cf. A156308, A127677, A217476, A263916.
%K A152060 tabl,sign
%O A152060 0,3
%A A152060 _Gary W. Adamson_ & _Roger L. Bagula_, Nov 22 2008
%E A152060 Edited by _L. Edson Jeffery_, Mar 26 2011
%E A152060 Some terms corrected from _Peter Bala_, May 04 2014