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A117165 Triangle of coefficients for the Shift-Moebius 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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,...].

LINKS

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

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.

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+n-i)%(c+n-i)==0, moebius((r+n-i)/(c+n-i)), 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

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Last modified August 28 15:16 EDT 2015. Contains 261125 sequences.