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A297742
Coefficients of polynomial whose zeros are the Möbius function.
0
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}
}
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
KEYWORD
sign,tabl
AUTHOR
Mats Granvik, Jan 05 2018
STATUS
approved