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A152060
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Triangle read by rows, characteristic polynomials of Cartan ring matrices.
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0
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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, 8, -16, 104, -352, 660, -672, -64, 4, 9, -18, 135, -546, 1287, -1782, 1386, -540, 81, -4, 1, -20, 170, -800, 2275, -4004, 4290
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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REFERENCES
| William G. Harter, University of Arkansas; personal communication
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LINKS
| Todd Rowland, Eric Weisstein's World of Mathematics, Cartan Matrix
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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, March 27, 2011.
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EXAMPLE
| 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, 64, -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].
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CROSSREFS
| Sequence in context: A093966 A103406 A142978 * A093190 A132191 A094437
Adjacent sequences: A152057 A152058 A152059 * A152061 A152062 A152063
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KEYWORD
| tabl,sign
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AUTHOR
| Gary W. Adamson & Roger L. Bagula (qntmpkt(AT)yahoo.com), Nov 22 2008
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EXTENSIONS
| Edited by L. Edson Jeffery, Mar 26 2011.
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