A282938 Recursive 2-parameter sequence allowing calculation of the Möbius function (not the same as A266378).

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

Offset: 1

Author

Gevorg Hmayakyan, Feb 25 2017

Keywords

• sign
• tabf

Comments

The a(n,m) forms a table where each row has (n-1)*(n-2)/2+1 = A000124(n-2) elements.

The index of the first row is n=1 and the index of the first column is m=0.

The right diagonal a(n, A000217(n-2)) = A008683(n), Möbius numbers, for n>=1.

Examples

			The first few rows starting from 1 follow:
  1
  -1
  1, -1
  -1, 2, -1, 0
  1, -2, 1, 0, -1, 2, -1
  -1, 3, -2, -1, 1, -1, 1, 1, 0, -2, 1
  1, -2, 1, 0, -1, 1, -1, 3, -1, 0, -1, -2, 1, 1, 1, -1
  -1, 3, -2, -1, 1, -1, 2, -2, 1, 1, 0, 0, -1, -3, 2, 2, 0, 0, -2, 1, 0, 0
		

Crossrefs

Cf. A000124, A000217, A008683, A231599.

Programs

Mathematica

nu[n_]:=(n-1)*(n-2)/2
U[n_, m_] := U[n, m] = If[n > 1, U[n - 1, m - n + 1] - U[n - 1, m], 0]
U[1, m_] := U[1, m] = If[m == 0, 1, 0]
a[n_, m_] := a[n, m] = If[(m < 0) || (nu[n] < m), 0, a[n - 1, m - n + 1] - a[n - 1, m] - a[n - 1, nu[n - 1]]*U[n - 1, m - 1]]
a[1, m_] := a[1, m] = If[m == 0, 1, 0]
Table[Table[a[n, m], {m, 0, nu[n]}], {n, 1, 30}]
Table[a[n, nu[n]], {n, 1, 50}]

Formula

a(n,m) = a(n-1, m-n+1) - a(n-1, m) - a(n-1, nu(n-1))*U(n-1, m-1),

where U(n,m) are coefficients of A231599, nu(n)=(n-1)*(n-2)/2, a(1,0)=1, a(n,m)=0 if m<0 and m>nu(n).

Möbius(n) = a(n,nu(n)).