A297742 - OEIS

%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