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

1, 1, -2, 1, -4, 3, 1, -6, 9, -4, 1, -8, 20, -16, 4, 1, -10, 35, -50, 25, -4, 1, -12, 54, -112, 105, -36, 4, 1, -14, 77, -210, 294, -196, 49, -4, 1, -16, 104, -352, 660, -672, 336, -64, 4, 1, -18, 135, -546, 1287, -1782, 1386, -540, 81, -4, 1, -20, 170, -800, 2275, -4004, 4290, -2640, 825, -100, 4

Offset: 0

Author

Gary W. Adamson & Roger L. Bagula, Nov 22 2008

Keywords

• tabl
• sign

Examples

			Triangle begins:
1;
1, -2;
1, -4, 3;
1, -6, 9, -4;
1, -8, 20, -16, 4;
1, -10, 35, -50, 25, -4;
1, -12, 54, -112, 105, -36, 4;
1, -14, 77, -210, 294, -196, 49, -4;
1, -16, 104, -352, 660, -672, 336, -64, 4;
1, -18, 135, -546, 1287, -1782, 1386, -540, 81, -4;
1, -20, 170, -800, 2275, -4004, 4290, -2640, 825, -100, 4;
...
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
[ 2,-1, 0, 0, 1]
[-1, 2,-1, 0, 0]
[ 0,-1, 2,-1, 0]
[ 0, 0,-1, 2,-1]
[ 1, 0, 0,-1, 2].
		

References

• William G. Harter, University of Arkansas; personal communication

Links

• P. Damianou , On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
• P. Damianou and C. Evripidou, Characteristic and Coxeter polynomials for affine Lie algebras, arXiv preprint arXiv:1409.3956 [math.RT], 2014.
• Todd Rowland, Eric Weisstein's World of Mathematics, Cartan Matrix

Crossrefs

Cf. A008310.

Cf. A156308, A127677, A217476, A263916.

Programs

Mathematica

M[n_] := SparseArray[{Band[{1, 1}] -> 2, Band[{1, 2}] -> -1, Band[{2, 1}] -> -1, {1, n} -> 1, {n, 1} -> 1}, {n, n}];
row[0] = {1}; row[1] = {1, -2};
row[n_] := (-1)^n CharacteristicPolynomial[M[n], x] // CoefficientList[#, x]& // Reverse;
Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Aug 08 2018 *)

Formula

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).

The coefficients of characteristic polynomials of matrices C_n, defined by

C_n=

(2 -1 0 ... 0 1)

(-1 2 -1 0 ... 0)

(0 -1 2 -1 0 ... 0)

...

(0 ... 0 -1 2 -1)

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

give the same triangle T(n,k), for n>0, k=0,...,n, with T(0,0)=1. - L. Edson Jeffery, Mar 27 2011

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

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

Extensions

Edited by L. Edson Jeffery, Mar 26 2011

Some terms corrected from Peter Bala, May 04 2014