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 A140882 A set of Cartan-like matrices with the properties that either the rows or columns as sums are zero that give a triangle of coefficients of characteristic polynomials: Example 3 X 3 matrix (row sums zero): {{2, -2, 0}, {-1, 2, -1}, {0, -2, 2}}. 2
 1, 2, -1, 0, -4, 1, 0, -8, 6, -1, 0, -12, 19, -8, 1, 0, -16, 44, -34, 10, -1, 0, -20, 85, -104, 53, -12, 1, 0, -24, 146, -259, 200, -76, 14, -1, 0, -28, 231, -560, 606, -340, 103, -16, 1, 0, -32, 344, -1092, 1572, -1210, 532, -134, 18, -1, 0, -36, 489, -1968, 3630, -3652, 2171, -784, 169, -20, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Row sums are: {1, 1, -3, -3, 0, 3, 3, 0, -3, -3, 0}. This sequence of matrices was inspired by the Kemeny "dominant", "hybrid" and "recessive" matrix of genetic characteristics: {{2, 2, 0}, {1, 2, 1}, {0, 2, 2}}/2^2. That type of matrix has row sums equal to one. I noticed that it resembled an unsigned Cartan matrix of a D_n or B_n type with rows sums zero. REFERENCES Kemeny, Snell and Thompson, Introduction to Finite Mathematics, 1966, Prentice-Hall, New Jersey, Section 3, Chapter VII, page 407. LINKS Table of n, a(n) for n=1..66. Pentti Haukkanen, Jorma Merikoski, Seppo Mustonen, Some polynomials associated with regular polygons, Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 178-193. FORMULA m(d) = If[ n == m, 2, If[(n == d &&m == d - 1) || (n == 1 && m == 2), -2, If[(n == m - 1 || n == m + 1), -1, 0]]; out_n,m=Coefficients(CharacteristicPolynomial(m(n)). Starting with the third row (0,-4,1), these coefficients appear to occur in the odd row polynomials in inverse powers of u for the o.g.f. (u-4)/(u-ux+x^2) of A267633, which generates a Fibonacci-type running average of adjacent polynomials of the o.g.f. involving these polynomials and others embedded in A228785. - Tom Copeland, Jan 16 2016 EXAMPLE 1; 2, -1; 0, -4, 1; 0, -8, 6, -1; 0, -12, 19, -8, 1; 0, -16, 44, -34, 10, -1; 0, -20, 85, -104, 53, -12, 1; 0, -24, 146, -259, 200, -76, 14, -1; 0, -28, 231, -560, 606, -340, 103, -16, 1; 0, -32, 344, -1092, 1572, -1210, 532, -134, 18, -1; 0, -36, 489, -1968, 3630, -3652, 2171, -784, 169, -20, 1; ... MATHEMATICA T[n_, m_, d_] := If[ n == m, 2, If[(n == d && m == d - 1) || ( n == 1 && m == 2), -2, If[(n == m - 1 || n == m + 1), -1, 0]]]; M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; a = Join[{{1}}, Table[CoefficientList[Det[M[ d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a] CROSSREFS Cf. A228785, A267633. Sequence in context: A185740 A139360 A326759 * A334044 A143724 A143425 Adjacent sequences: A140879 A140880 A140881 * A140883 A140884 A140885 KEYWORD uned,tabl,sign AUTHOR Roger L. Bagula and Gary W. Adamson, Jul 22 2008 STATUS approved

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Last modified December 7 23:30 EST 2023. Contains 367662 sequences. (Running on oeis4.)