

A117165


Triangle of coefficients for the ShiftMoebius transform, read by rows.


10



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
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OFFSET

1,4


COMMENTS

Column k = ShiftMoebius transform of a sequence of all zeros except for a single '1' in position k: [0,0,0,..(k1)zeros..,1,0,0,0,...].
Column 1 is A117166, the ShiftMoebius transform of [1,0,0,0,...].
Column 2 is A117167, the ShiftMoebius transform of [0,1,0,0,...].
Column 3 is A117168, the ShiftMoebius transform of [0,0,1,0,...].
Row sums give A117169, the ShiftMoebius transform of [1,1,1,...].


LINKS

Table of n, a(n) for n=1..105.


FORMULA

The ShiftMoebius 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.


EXAMPLE

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; ...


PROG

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


CROSSREFS

Cf. A117166 (column 1), A117167 (column 2), A117168 (column 3), A117169 (row sums), A117170 (inverse), A117162, A008683; A117175.
Sequence in context: A101659 A074272 A105553 * A213629 A024363 A050600
Adjacent sequences: A117162 A117163 A117164 * A117166 A117167 A117168


KEYWORD

sign,tabl


AUTHOR

Wouter Meeussen and Paul D. Hanna, Mar 05 2006


STATUS

approved



