

A140882


A set of Cartanlike 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
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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, PrenticeHall, New Jersey, Section 3, Chapter VII, page 407.


LINKS



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. (u4)/(uux+x^2) of A267633, which generates a Fibonaccitype 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



KEYWORD



AUTHOR



STATUS

approved



