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Search: id:A101391
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| A101391 |
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Triangle read by rows: T(n,k) is the number of compositions of n into k relatively prime summands (2<=k<=n). |
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1
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| 1, 2, 1, 2, 3, 1, 4, 6, 4, 1, 2, 9, 10, 5, 1, 6, 15, 20, 15, 6, 1, 4, 18, 34, 35, 21, 7, 1, 6, 27, 56, 70, 56, 28, 8, 1, 4, 30, 80, 125, 126, 84, 36, 9, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 4, 42, 154, 325, 461, 462, 330, 165, 55, 11, 1, 12, 66, 220, 495, 792, 924, 792
(list; table; graph; listen)
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OFFSET
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2,2
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COMMENT
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Mirror image of A039911 Sum of entries in row n = A000740(n). Column 2 yields A000010 (phi(n)). Column 3 yields A000741. Column 4 yields A000742. Column 5 yields A000743. Column 6 yields A023031. Column 7 yields A023032. Column 8 yields A023033. Column 9 yields A023034.
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REFERENCES
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H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2 (1964), 241-260.
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FORMULA
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T(n, k)=sum_{d|n}(binomial(d-1, k-1)*mobius(n/d).
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EXAMPLE
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T(6,3)=9 because we have 411,141,114, and the six permutations of 123 (222 does not qualify).
T(8,3)=18 because binomial(0,2)*mobius(8/1)+binomial(1,2)*mobius(8/2)+binomial(3,2)*mobius(8/4)+binomial(7,2)*mobius(8/8)=0+0+(-3)+21=18.
Triangle begins:
1;
2,1;
2,3,1;
4,6,4,1;
2,9,10,5,1;
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MAPLE
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with(numtheory): T:=proc(n, k) local d, j, b: d:=divisors(n): for j from 1 to tau(n) do b[j]:=binomial(d[j]-1, k-1)*mobius(n/d[j]) od: sum(b[i], i=1..tau(n)) end: for n from 2 to 14 do seq(T(n, k), k=2..n) od; # yields the sequence in triangular form
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CROSSREFS
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Cf. A039911, A000740, A000010, A000741, A000742, A000743, A023031, A023032, A023033, A023034.
Sequence in context: A055884 A055889 A125930 this_sequence A117704 A078032 A008313
Adjacent sequences: A101388 A101389 A101390 this_sequence A101392 A101393 A101394
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 26 2005
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