login
Recursive 2-parameter sequence allowing calculation of the Möbius function (not the same as A266378)
0

%I #12 Nov 22 2024 00:35:28

%S 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,

%T 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,

%U 2,2,0,0,-2,1,0,0,1,-3,2,1,-1,1,-2,2,-2,2,-2,-1

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

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

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

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

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

%F 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).

%F Möbius(n) = a(n,nu(n)).

%e The first few rows starting from 1 follow:

%e 1

%e -1

%e 1, -1

%e -1, 2, -1, 0

%e 1, -2, 1, 0, -1, 2, -1

%e -1, 3, -2, -1, 1, -1, 1, 1, 0, -2, 1

%e 1, -2, 1, 0, -1, 1, -1, 3, -1, 0, -1, -2, 1, 1, 1, -1

%e -1, 3, -2, -1, 1, -1, 2, -2, 1, 1, 0, 0, -1, -3, 2, 2, 0, 0, -2, 1, 0, 0

%t nu[n_]:=(n-1)*(n-2)/2

%t U[n_, m_] := U[n, m] = If[n > 1, U[n - 1, m - n + 1] - U[n - 1, m], 0]

%t U[1, m_] := U[1, m] = If[m == 0, 1, 0]

%t 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]]

%t a[1, m_] := a[1, m] = If[m == 0, 1, 0]

%t Table[Table[a[n, m], {m, 0, nu[n]}], {n, 1, 30}]

%t Table[a[n, nu[n]], {n, 1, 50}]

%Y Cf. A000124, A000217, A008683, A231599.

%K sign,tabf,changed

%O 1,6

%A _Gevorg Hmayakyan_, Feb 25 2017