A157972 Coefficients of the even n Cartan A_n characteristic polynomials factored modulo two: m(n,m,d)=If[ n == m, 2, If[n == m - 1 || n == m + 1, -1, 0]].

1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1

Offset

2,1

Comments

Row sums are:

{2, 3, 3, 2, 5, 5, 2, 3, 7, 2,...},

The interesting effect is that the even terms are squares, and every third of them has (x+1).

Links

Table of n, a(n) for n=2..44.

Formula

m(n,m,d)=If[ n == m, 2, If[n == m - 1 || n == m + 1, -1, 0]];

p(x,n)=Characteristicpolynomial(m(n,m,k),x);

q(x,2*n,i)=Factor(p(x,2*n),i);

t(n,m)=coefficients(q(x,2*n,1))

Example

Examples of polynomials (all):
x,
(1 + x)^2,
x^3,
(1 + x + x^2)^2,
x( 1 + x)^4,
(1 + x^2 + x^3)^2,
x^7,
(1 + x)^2(1 + x + x^3)^2,
x(1 + x + x^2)^4,
(1 + x + x^2 + x^4 + x^5)^2
Coefficients even first polynomials:
{1, 1},
{1, 1, 1},
{1, 0, 1, 1},
{1, 1},
{1, 1, 1, 0, 1, 1},
{1, 1, 0, 0, 1, 1, 1},
{1, 1},
{1, 1, 0, 0, 1},
{1, 1, 0, 0, 1, 1, 1, 0, 1, 1},
{1, 1}

Mathematica

Clear[M, T, d, a, x, a0];
T[n_, m_, d_] := If[ n == m, 2, If[n == m - 1 || n == m + 1, -1, 0]];
M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}];
Table[FactorList[CharacteristicPolynomial[M[d], x], Modulus -> 2][[2]][[1]], {d, 2, 20, 2}];
Table[CoefficientList[FactorList[CharacteristicPolynomial[M[d], x], Modulus -> 2][[2]][[1]], x], {d, 2, 20, 2}];
Flatten[%]
Table[Apply[Plus, CoefficientList[FactorList[CharacteristicPolynomial[M[ d], x], Modulus -> 2][[2]][[1]], x]], {d, 2, 20, 2}];

Crossrefs

Sequence in context: A249133 A118102 A089509 * A124897 A071035 A138345

Adjacent sequences: A157969 A157970 A157971 * A157973 A157974 A157975

Keyword

nonn,tabl

Author

Roger L. Bagula and Gary W. Adamson, Mar 10 2009

Status

approved