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}}.

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

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