cp's OEIS Frontend

A143809 Eigentriangle of the Mobius transform, (A054525).

%I A143809 #6 Nov 20 2022 10:53:14
%S A143809 1,-1,1,-1,0,0,0,-1,0,-1,-1,0,0,0,-2,1,-1,0,0,0,-3,-1,0,0,0,0,0,-3,0,
%T A143809 0,0,1,0,0,0,-4,0,0,0,0,0,0,0,0,-3,1,-1,0,0,2,0,0,0,0,-3,-1,0,0,0,0,0,
%U A143809 0,0,0,0,-1,0,1,0,1,0,3,0,0,0,0,0,-2,-1,0,0,0,0,0,0,0,0,0,0,0,3,1,-1,0,0,0
%N A143809 Eigentriangle of the Mobius transform, (A054525).
%C A143809 The eigentriangle of the Mobius transform may be defined by the operation consisting of the termwise product of A054525 row terms and the first n terms of A007554, where A007554: (1, 1, 0, -1, -2, -3, -3,...) = the eigensequence of A054525.
%C A143809 This triangle has the following properties:
%C A143809 Sum of n-th row terms = rightmost term of next row.
%C A143809 Right border = A007554, the eigensequence of the Mobius transform.
%C A143809 Row sums = A007554 shifted one place to the left: (1, 0, -1, -2, -3,...).
%C A143809 Left border = mu(n), A008683.
%C A143809 A054525 = the Mobius transform and A007554 = the eigensequence of A054525.
%F A143809 Triangle read by rows, A054525 * (A007554 * 0^(n-k)); 1<=k<=n
%e A143809 First few rows of the triangle:
%e A143809    1;
%e A143809   -1,  1;
%e A143809   -1,  0, 0;
%e A143809    0, -1, 0, -1;
%e A143809   -1,  0, 0,  0, -2;
%e A143809    1, -1, 0,  0,  0, -3;
%e A143809   -1,  0, 0,  0,  0,  0, -3;
%e A143809    0,  0, 0,  1,  0,  0,  0, -4;
%e A143809    0,  0, 0,  0,  0,  0,  0,  0, -3;
%e A143809    1, -1, 0,  0,  2,  0,  0,  0,  0, -3;
%e A143809    ...
%e A143809 Row 6 = (1, -1, 0, 0, 0, -3) = termwise product of row 6 of the Mobius transform (1, -1, -1, 0, 0, 1) and the first 6 terms of A007554, (the eigensequence of the Mobius transform): (1, 1, 0, -1, -2, -3).
%Y A143809 Cf. A008683, A007554, A054525.
%K A143809 tabl,sign
%O A143809 1,15
%A A143809 _Gary W. Adamson_, Sep 01 2008