A117163 -id:A117163 - cp's OEIS frontend

A117164 Column 3 of triangle A117162; equals the Moebius transform of A117163 (column 2 of A117162) preceded by a zero.

0, 0, 1, -1, -2, -1, -1, 2, -1, 3, -1, 2, 2, 1, -2, 1, 2, 1, -1, 0, 1, 1, -1, -2, 2, -2, -1, 2, 3, 1, -1, -3, -2, -2, 3, -1, 2, 1, -5, 0, 2, -1, 2, 0, 4, -1, -2, -1, -1, -4, -3, 2, 3, 1, 2, -4, 1, -2, 0, 0, -1, 1, -1, 1, 1, 2, 2, 0, 0, -4, 3, -1, -2, -2, -1, 1, 5, 3, 1, -4, -3, -3, -1, -2, -3, -2, -3, 1, 2, -4, 1, 2, 3, 4, 2, 3, 4, 1, 4

Offset: 1

Wouter Meeussen and Paul D. Hanna, Mar 05 2006

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The inverse Moebius transform of column 4 of A117162 equals

this sequence preceded by a zero.

Cf. A117162, A112682, A008683 (Moebius); A117163 (column 2).

Equals the Moebius transform of column 2 preceded by a zero:

[0,0,1,-1,-2,-1,-1,2,...] = Moebius([0, 0,1,-1,-2,0,-1,1,...]).

A117162 Matrix inverse of triangle A112682.

Original entry on oeis.org

1, -1, 1, -1, -1, 1, 0, -2, -1, 1, -1, 0, -2, -1, 1, 1, -1, -1, -2, -1, 1, -1, 1, -1, -1, -2, -1, 1, 0, 0, 2, -2, -1, -2, -1, 1, 0, 1, -1, 2, -2, -1, -2, -1, 1, 1, -1, 3, 0, 1, -2, -1, -2, -1, 1, -1, 1, -1, 3, 0, 1, -2, -1, -2, -1, 1, 0, 2, 2, 0, 4, -1, 1, -2, -1, -2, -1, 1, -1, 0, 2, 2, 0, 4, -1, 1, -2, -1, -2, -1, 1

Offset: 1

Wouter Meeussen and Paul D. Hanna, Mar 05 2006

The limit of the columns (without leading zeros) is A117166, the Shift-Moebius transform of [1,0,0,0,...] (cf. A117165).

			Column 1 equals A008683 = Moebius transform of [1,0,0,0,...].
Column 2 = Moebius transform of column 1 preceded by a zero: [0,1,-1,-2,0,-1,1,0,...] = Moebius([0, 1,-1,-1,0,-1,1,-1,...]).
Column 3 = Moebius transform of column 2 preceded by a zero: [0,0,1,-1,-2,-1,-1,2,...] = Moebius([0, 0,1,-1,-2,0,-1,1,...]).
Column 4 = Moebius transform of column 3 preceded by a zero: [0,0,0,1,-1,-2,-1,-2,...] = Moebius([0, 0,0,1,-1,-2,-1,-1,...]).
Triangle begins:
1;
-1, 1;
-1,-1, 1;
0,-2,-1, 1;
-1, 0,-2,-1, 1;
1,-1,-1,-2,-1, 1;
-1, 1,-1,-1,-2,-1, 1;
0, 0, 2,-2,-1,-2,-1, 1;
0, 1,-1, 2,-2,-1,-2,-1, 1;
1,-1, 3, 0, 1,-2,-1,-2,-1, 1;
-1, 1,-1, 3, 0, 1,-2,-1,-2,-1, 1;
0, 2, 2, 0, 4,-1, 1,-2,-1,-2,-1, 1;
-1, 0, 2, 2, 0, 4,-1, 1,-2,-1,-2,-1, 1; ...

Cf. A112682 (inverse), A008683 (column 1), A117163 (column 2), A117164 (column 3); A117165 (Shift-Moebius), A117170 (inverse Shift-Moebius).

Column k+1 equals the Moebius transform of column k preceded by a zero, where column k includes the k-1 zeros above the diagonal, for k>=1, starting with A008683 in column 1.

cp's OEIS Frontend

A117164 Column 3 of triangle A117162; equals the Moebius transform of A117163 (column 2 of A117162) preceded by a zero.

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