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%I #21 Feb 03 2018 12:38:59
%S 0,1,-1,-1,0,1,1,1,-1,-1,0,-1,-1,1,1,0,1,2,0,-2,-1,0,1,1,-2,-2,1,1,0,
%T -1,-2,1,4,1,-2,-1,0,0,1,2,-1,-4,-1,2,1,0,0,0,-1,-2,1,4,1,-2,-1,0,0,0,
%U -1,-1,3,3,-3,-3,1,1,0,0,0,1,2,-2,-6,0,6,2,-2,-1
%N Coefficients of polynomial whose zeros are the Möbius function.
%C Also determinant polynomial whose roots are the Möbius function A008683, see formula section.
%C The table (A054431 - x*A051731) starts:
%C {
%C {1 - x, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
%C {1 - x, -x, 1, 0, 1, 0, 1, 0, 1, 0, 1},
%C {1 - x, 1, -x, 1, 1, 0, 1, 1, 0, 1, 1},
%C {1 - x, -x, 1, -x, 1, 0, 1, 0, 1, 0, 1},
%C {1 - x, 1, 1, 1, -x, 1, 1, 1, 1, 0, 1},
%C {1 - x, -x, -x, 0, 1, -x, 1, 0, 0, 0, 1},
%C {1 - x, 1, 1, 1, 1, 1, -x, 1, 1, 1, 1},
%C {1 - x, -x, 1, -x, 1, 0, 1, -x, 1, 0, 1},
%C {1 - x, 1, -x, 1, 1, 0, 1, 1, -x, 1, 1},
%C {1 - x, -x, 1, 0, -x, 0, 1, 0, 1, -x, 1},
%C {1 - x, 1, 1, 1, 1, 1, 1, 1, 1, 1, -x}
%C }
%F Let A be the lower triangular matrix: if n mod k = 0 then 1 else 0.
%F Let B the upper triangular matrix: if k mod n = 0 then A008683(n) else 0.
%F The polynomial is then: determinant(A.B - x*A) where . stands for matrix multiplication and * stands for normal multiplication like 2*3=6. x is the variable to solve for: polynomial = determinant(A054431 - x*A051731).
%e The table of polynomial coefficients starts:
%e {
%e { 0},
%e { 1, -1},
%e {-1, 0, 1},
%e { 1, 1, -1, -1},
%e { 0, -1, -1, 1, 1},
%e { 0, 1, 2, 0, -2, -1},
%e { 0, 1, 1, -2, -2, 1, 1},
%e { 0, -1, -2, 1, 4, 1, -2, -1},
%e { 0, 0, 1, 2, -1, -4, -1, 2, 1},
%e { 0, 0, 0, -1, -2, 1, 4, 1, -2, -1},
%e { 0, 0, 0, -1, -1, 3, 3, -3, -3, 1, 1},
%e { 0, 0, 0, 1, 2, -2, -6, 0, 6, 2, -2, -1}
%e }
%t (* program 1 *)
%t Clear[x, P]
%t TableForm[polynomial = Table[
%t A = Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, nn}], {n, 1, nn}];
%t B = Table[
%t Table[If[Mod[k, n] == 0, MoebiusMu[n], 0], {k, 1, nn}], {n, 1,
%t nn}];
%t Det[A.B - x*A], {nn, 1, 11}]];
%t Flatten[CoefficientList[polynomial, x]]
%Y Cf. A008683, A051731, A054431.
%K sign,tabl
%O 0,18
%A _Mats Granvik_, Jan 05 2018
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