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A077049
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Left summatory matrix, T, by antidiagonals.
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12
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1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| If S=(s(1),s(2),...) is a sequence written as a column vector, then T*S is the summatory sequence of S; i.e. its n-th term is Sum{s(k): k|n}. T is the inverse of the left Moebius transformation matrix, A077050. Except for the first term in some cases, Column 1 of T^(-2) is A007427, Column 1 of T^(-1) is A008683, Column 1 of T^2 is A000005, Column 1 of T^3 is A007425.
This is essentially the same as A051731, which includes only the triangle. Note that the standard in the OEIS is left to right antidiagonals, which would make this the right summatory matrix, and A077051 the left one. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Apr 08 2009]
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 28 2010: (Start)
As defined with antidiagonals of the array = the triangle shown in the
example section. Row sums of this triangle = A032741 (with a different
offset): 1, 1, 2, 1, 3, 1, 3,...
Let the triangle = M. Then Lim_{n-inf} M^n = A002033, the left-shifted
vector considered as a sequence: (1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8,...). (End)
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LINKS
| C. Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.
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FORMULA
| T(n, k)=1 if k|n, else T(n, k)=0, k>=1, n>=1.
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EXAMPLE
| T(4,2)=1 since 2 divides 4. Northwest corner:
1 0 0 0 0 0
1 1 0 0 0 0
1 0 1 0 0 0
1 1 0 1 0 0
1 0 0 0 1 0
1 1 1 0 0 1
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 28 2010: (Start)
First few rows of the triangle =
1;
1, 0;
1, 1, 0;
1, 0, 0, 0;
1, 1, 1, 0, 0;
1, 0, 0, 0, 0, 0;
1, 1, 0, 1, 0, 0, 0;
1, 0, 1, 0, 0, 0, 0, 0;
1, 1, 0, 0, 1, 0, 0, 0, 0;
... (End)
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CROSSREFS
| Cf. A077050, A077051, A077052.
Cf. A032741, A002033 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 28 2010]
Sequence in context: A014577 A157926 A131377 * A124895 A089885 A143242
Adjacent sequences: A077046 A077047 A077048 * A077050 A077051 A077052
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KEYWORD
| nonn,tabl
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu), Oct 22 2002
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