A117165 Triangle of coefficients for the Shift-Moebius transform, read by rows.

1, -1, 1, -2, 0, 1, -1, -1, 0, 1, -2, -1, 0, 0, 1, 1, -2, -1, 0, 0, 1, -1, -1, -1, 0, 0, 0, 1, 3, -2, -1, -1, 0, 0, 0, 1, 0, 0, -2, -1, 0, 0, 0, 0, 1, 4, -2, -1, -1, -1, 0, 0, 0, 0, 1, 4, 0, -2, -1, -1, 0, 0, 0, 0, 0, 1, 5, 1, -1, -2, -1, -1, 0, 0, 0, 0, 0, 1, 1, 2, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 1, 7, 0, 0, -2, -1, -1, -1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Wouter Meeussen and Paul D. Hanna, Mar 05 2006

Keywords

• sign
• tabl

Comments

Column k = Shift-Moebius transform of a sequence of all zeros except for a single '1' in position k: [0,0,0,..(k-1)zeros..,1,0,0,0,...].

Column 1 is A117166, the Shift-Moebius transform of [1,0,0,0,...].

Column 2 is A117167, the Shift-Moebius transform of [0,1,0,0,...].

Column 3 is A117168, the Shift-Moebius transform of [0,0,1,0,...].

Row sums give A117169, the Shift-Moebius transform of [1,1,1,...].

Examples

			Triangle begins:
1;
-1, 1;
-2, 0, 1;
-1,-1, 0, 1;
-2,-1, 0, 0, 1;
1,-2,-1, 0, 0, 1;
-1,-1,-1, 0, 0, 0, 1;
3,-2,-1,-1, 0, 0, 0, 1;
0, 0,-2,-1, 0, 0, 0, 0, 1;
4,-2,-1,-1,-1, 0, 0, 0, 0, 1;
4, 0,-2,-1,-1, 0, 0, 0, 0, 0, 1;
5, 1,-1,-2,-1,-1, 0, 0, 0, 0, 0, 1;
1, 2,-1,-1,-1,-1, 0, 0, 0, 0, 0, 0, 1;
7, 0, 0,-2,-1,-1,-1, 0, 0, 0, 0, 0, 0, 1;
6, 3,-2,-1,-2,-1,-1, 0, 0, 0, 0, 0, 0, 0, 1;
5, 3, 1,-2,-1,-1,-1,-1, 0, 0, 0, 0, 0, 0, 0, 1; ...
		

Crossrefs

Cf. A117166 (column 1), A117167 (column 2), A117168 (column 3), A117169 (row sums), A117170 (inverse), A117162, A008683; A117175.

Programs

PARI

{T(n,k)=if(n=c,if((r+n-i)%(c+n-i)==0,moebius((r+n-i)/(c+n-i)),0))))[ n,k])}

Formula

The Shift-Moebius transform of a sequence B is equal to the limit of the iteration: let C_1 = B and for k>1, C_{k+1} = Moebius transform of C_k preceded by k zeros, then shift left k places (to drop the leading k zeros).

Triangle A117162 is a good example, starting with A008683 in column 1 as C_1 and each column k, C_k, is obtained using the above iteration, so that the columns converge to A117166.