%I A157972
%S A157972 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,
%T A157972 1,1,1,0,1,1,1,1
%N 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]].
%C A157972 Row sums are:
%C A157972 {2, 3, 3, 2, 5, 5, 2, 3, 7, 2,...},
%C A157972 The interesting effect is that the even terms are squares, and every 
               third of them has (x+1).
%F A157972 m(n,m,d)=If[ n == m, 2, If[n == m - 1 || n == m + 1, -1, 0]];
%F A157972 p(x,n)=Characteristicpolynomial(m(n,m,k),x);
%F A157972 q(x,2*n,i)=Factor(p(x,2*n),i);
%F A157972 t(n,m)=coefficients(q(x,2*n,1))
%e A157972 Examples of polynomials (all):
%e A157972 x,
%e A157972 (1 + x)^2,
%e A157972 x^3,
%e A157972 (1 + x + x^2)^2,
%e A157972 x( 1 + x)^4,
%e A157972 (1 + x^2 + x^3)^2,
%e A157972 x^7,
%e A157972 (1 + x)^2(1 + x + x^3)^2,
%e A157972 x(1 + x + x^2)^4,
%e A157972 (1 + x + x^2 + x^4 + x^5)^2
%e A157972 Coefficients even first polynomials:
%e A157972 {1, 1},
%e A157972 {1, 1, 1},
%e A157972 {1, 0, 1, 1},
%e A157972 {1, 1},
%e A157972 {1, 1, 1, 0, 1, 1},
%e A157972 {1, 1, 0, 0, 1, 1, 1},
%e A157972 {1, 1},
%e A157972 {1, 1, 0, 0, 1},
%e A157972 {1, 1, 0, 0, 1, 1, 1, 0, 1, 1},
%e A157972 {1, 1}
%t A157972 Clear[M, T, d, a, x, a0];
%t A157972 T[n_, m_, d_] := If[ n == m, 2, If[n == m - 1 || n == m + 1, -1, 0]];
%t A157972 M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}];
%t A157972 Table[FactorList[CharacteristicPolynomial[M[d], x], Modulus -> 2][[2]][[1]], 
               {d, 2, 20, 2}];
%t A157972 Table[CoefficientList[FactorList[CharacteristicPolynomial[M[d], x], Modulus 
               -> 2][[2]][[1]], x], {d, 2, 20, 2}];
%t A157972 Flatten[%]
%t A157972 Table[Apply[Plus, CoefficientList[FactorList[CharacteristicPolynomial[M[ 
               d], x], Modulus -> 2][[2]][[1]], x]], {d, 2, 20, 2}];
%Y A157972 Sequence in context: A152904 A118102 A089509 this_sequence A124897 A138345 
               A014590
%Y A157972 Adjacent sequences: A157969 A157970 A157971 this_sequence A157973 A157974 
               A157975
%K A157972 nonn,tabl
%O A157972 2,1
%A A157972 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Mar 
               10 2009