A297742 Coefficients of polynomial whose zeros are the Möbius function.

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, -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, -1, -1, 3, 3, -3, -3, 1, 1, 0, 0, 0, 1, 2, -2, -6, 0, 6, 2, -2, -1

Offset

0,18

Comments

Also determinant polynomial whose roots are the Möbius function A008683, see formula section.

The table (A054431 - x*A051731) starts:

{

{1 - x, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},

{1 - x, -x, 1, 0, 1, 0, 1, 0, 1, 0, 1},

{1 - x, 1, -x, 1, 1, 0, 1, 1, 0, 1, 1},

{1 - x, -x, 1, -x, 1, 0, 1, 0, 1, 0, 1},

{1 - x, 1, 1, 1, -x, 1, 1, 1, 1, 0, 1},

{1 - x, -x, -x, 0, 1, -x, 1, 0, 0, 0, 1},

{1 - x, 1, 1, 1, 1, 1, -x, 1, 1, 1, 1},

{1 - x, -x, 1, -x, 1, 0, 1, -x, 1, 0, 1},

{1 - x, 1, -x, 1, 1, 0, 1, 1, -x, 1, 1},

{1 - x, -x, 1, 0, -x, 0, 1, 0, 1, -x, 1},

{1 - x, 1, 1, 1, 1, 1, 1, 1, 1, 1, -x}

}

Links

Table of n, a(n) for n=0..77.

Formula

Let A be the lower triangular matrix: if n mod k = 0 then 1 else 0.

Let B the upper triangular matrix: if k mod n = 0 then A008683(n) else 0.

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).

Example

The table of polynomial coefficients starts:
{
  { 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, -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, -1, -1,  3,  3, -3, -3,  1,  1},
  { 0,  0,  0,  1,  2, -2, -6,  0,  6,  2, -2, -1}
}

Mathematica

(* program 1 *)
Clear[x, P]
TableForm[polynomial = Table[
    A = Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, nn}], {n, 1, nn}];
    B = Table[
      Table[If[Mod[k, n] == 0, MoebiusMu[n], 0], {k, 1, nn}], {n, 1,
       nn}];
    Det[A.B - x*A], {nn, 1, 11}]];
Flatten[CoefficientList[polynomial, x]]

Crossrefs

Cf. A008683, A051731, A054431.

Sequence in context: A198393 A083054 A336921 * A061264 A193680 A186809

Adjacent sequences: A297739 A297740 A297741 * A297743 A297744 A297745

Keyword

sign,tabl

Author

Mats Granvik, Jan 05 2018

Status

approved