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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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2
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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; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| 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
| Alois P. Heinz, Table of n, a(n) for n = 1..10011
Index entries for sequences related to Lyndon words
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FORMULA
| 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
| 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
| 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
| Columns 1-2: A000012, A085945. Rows 1-4: A000012, A000027, A028387, A003777. See also A143325, A143326, A134541, A008683.
Sequence in context: A049513 A121207 A097084 * A094954 A083064 A204057
Adjacent sequences: A143324 A143325 A143326 * A143328 A143329 A143330
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KEYWORD
| nonn,tabl
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AUTHOR
| Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 07 2008
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