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A117165
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Triangle of coefficients for the Shift-Moebius transform, read by rows.
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10
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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
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1,4
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COMMENTS
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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,...].
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LINKS
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FORMULA
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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.
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EXAMPLE
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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; ...
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PROG
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(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])}
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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