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Search: id:A143327
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| A143327 |
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Table T(n,k) by antidiagonals. T(n,k) is the number of primitive (=aperiodic) k-ary words (n,k >= 1) with length less or equal to n which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet. |
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+0
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| 1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 11, 1, 1, 5, 19, 35, 26, 1, 1, 6, 29, 79, 115, 53, 1, 1, 7, 41, 149, 334, 347, 116, 1, 1, 8, 55, 251, 773, 1339, 1075, 236, 1, 1, 9, 71, 391, 1546, 3869, 5434, 3235, 488, 1, 1, 10, 89, 575, 2791, 9281, 19493, 21754, 9787, 983, 1, 1, 11
(list; table; graph; listen)
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OFFSET
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1,5
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COMMENT
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The coefficients of the polynomial of row n are given by the n-th row of triangle A134541; for example row 4 has polynomial -1+k^2+k^3.
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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_{1<=j<=n} Sum_{d|j} k^(d-1)*mu(j/d). T(n, k) = Sum_{1<=j<=n} A143325(j, k). T(n, k) = A143326(n, k) / k.
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EXAMPLE
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T(3,3)=11, because 11 words of length <=3 over 3-letter alphabet {a,b,c} are primitive and earlier than others derived by cyclic shifts of the alphabet: a, ab, ac, aab, aac, aba, abb, abc, aca, acb, acc.
Table begins:
1 1 1 1 1 ...
1 2 3 4 5 ...
1 5 11 19 29 ...
1 11 35 79 149 ...
1 26 115 334 773 ...
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MAPLE
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with (numtheory): f1 := proc (n) option remember; unapply (k^(n-1)-add(f1(d)(k), d=divisors(n)minus{n}), k) end; g1 := proc (n) option remember; unapply ( add (f1(j)(x), j=1..n), x); end; T := (n, k) -> g1(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 1-2: A000012, A085945. Rows 1-4: A000012, A000027, A028387, A003777. See also A143325, A143326, A134541, A008683.
Sequence in context: A049513 A121207 A097084 this_sequence A094954 A083064 A112338
Adjacent sequences: A143324 A143325 A143326 this_sequence A143328 A143329 A143330
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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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