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

%I A282938 #14 Jun 02 2025 12:20:48
%S A282938 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 A282938 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 A282938 2,2,0,0,-2,1,0,0,1,-3,2,1,-1,1,-2,2,-2,2,-2,-1
%N A282938 Recursive 2-parameter sequence allowing calculation of the Möbius function (not the same as A266378).
%C A282938 The a(n,m) forms a table where each row has (n-1)*(n-2)/2+1 = A000124(n-2) elements.
%C A282938 The index of the first row is n=1 and the index of the first column is m=0.
%C A282938 The right diagonal a(n, A000217(n-2)) = A008683(n), Möbius numbers, for n>=1.
%F A282938 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 A282938 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 A282938 Möbius(n) = a(n,nu(n)).
%e A282938 The first few rows starting from 1 follow:
%e A282938   1
%e A282938   -1
%e A282938   1, -1
%e A282938   -1, 2, -1, 0
%e A282938   1, -2, 1, 0, -1, 2, -1
%e A282938   -1, 3, -2, -1, 1, -1, 1, 1, 0, -2, 1
%e A282938   1, -2, 1, 0, -1, 1, -1, 3, -1, 0, -1, -2, 1, 1, 1, -1
%e A282938   -1, 3, -2, -1, 1, -1, 2, -2, 1, 1, 0, 0, -1, -3, 2, 2, 0, 0, -2, 1, 0, 0
%t A282938 nu[n_]:=(n-1)*(n-2)/2
%t A282938 U[n_, m_] := U[n, m] = If[n > 1, U[n - 1, m - n + 1] - U[n - 1, m], 0]
%t A282938 U[1, m_] := U[1, m] = If[m == 0, 1, 0]
%t A282938 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 A282938 a[1, m_] := a[1, m] = If[m == 0, 1, 0]
%t A282938 Table[Table[a[n, m], {m, 0, nu[n]}], {n, 1, 30}]
%t A282938 Table[a[n, nu[n]], {n, 1, 50}]
%Y A282938 Cf. A000124, A000217, A008683, A231599.
%K A282938 sign,tabf
%O A282938 1,6
%A A282938 _Gevorg Hmayakyan_, Feb 25 2017