1,4

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,...].

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 a 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.

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; ...

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

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

Sequence in context: A101659 A074272 A105553 this_sequence A024363 A050600 A129691

Adjacent sequences: A117162 A117163 A117164 this_sequence A117166 A117167 A117168

sign,tabl

Wouter Meeussen (wouter.meeussen(AT)pandora.be) and Paul D. Hanna (pauldhanna(AT)juno.com), Mar 05 2006