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Search: id:A143324
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| A143324 |
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Table T(n,k) by antidiagonals. T(n,k) is the number of length n primitive (=aperiodic or period n) k-ary words (n,k >= 1). |
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4
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| 1, 2, 0, 3, 2, 0, 4, 6, 6, 0, 5, 12, 24, 12, 0, 6, 20, 60, 72, 30, 0, 7, 30, 120, 240, 240, 54, 0, 8, 42, 210, 600, 1020, 696, 126, 0, 9, 56, 336, 1260, 3120, 4020, 2184, 240, 0, 10, 72, 504, 2352, 7770, 15480, 16380, 6480, 504, 0, 11, 90, 720, 4032, 16800, 46410, 78120
(list; table; graph; listen)
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OFFSET
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1,2
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COMMENT
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Column k is Dirichlet convolution of mu(n) with k^n.
The coefficients of the polynomial of row n are given by the n-th row of triangle A054525; for example row 4 has polynomial -k^2+k^4.
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LINKS
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Index entries for sequences related to Lyndon words
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FORMULA
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T(n, k) = Sum_{d|n} k^d*mu(n/d). T(n, k) = k^n - Sum_{d<n,d|n} T(d,k). T(n, k) = A143325(n, k) * k. T(n, k) = A074650(n, k) * n.
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EXAMPLE
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T(2,3)=6, because there are 6 primitive words of length 2 over 3-letter alphabet {a,b,c}: ab, ac, ba, bc, ca, cb; note that the non-primitive words aa, bb and cc don't belong to the list; secondly note that the words in the list need not be Lyndon words, for example ba can be derived from ab by a cyclic rotation of the positions.
Table begins:
1 2 3 4 5 ...
0 2 6 12 20 ...
0 6 24 60 120 ...
0 12 72 240 600 ...
0 30 240 1020 3120 ...
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MAPLE
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with (numtheory): f0 := proc (n) option remember; unapply (k^n-add(f0(d)(k), d=divisors(n)minus{n}), k) end; T := (n, k)-> f0(n)(k); seq (seq(T(i, d-i), i=1..d-1), d=2..13);
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CROSSREFS
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Columns 1-6: A000007, A027375, A054718, A054719, A054720, A054721. Rows 1-5, 7: A000027, A002378, A007531, A047928, A061167, A133499. See also A074650, A143325, A008683, A054525.
Sequence in context: A141432 A115241 A154559 this_sequence A097418 A154752 A156776
Adjacent sequences: A143321 A143322 A143323 this_sequence A143325 A143326 A143327
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KEYWORD
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nonn,tabl
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AUTHOR
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Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 07 2008
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