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A157972
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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]].
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 2,1
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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).
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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))
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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}
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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}];
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CROSSREFS
| Sequence in context: A071033 A118102 A089509 * A124897 A071035 A138345
Adjacent sequences: A157969 A157970 A157971 * A157973 A157974 A157975
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KEYWORD
| nonn,tabl
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AUTHOR
| Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Mar 10 2009
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