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A143326
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Table T(n,k) by antidiagonals. T(n,k) is the number of primitive (=aperiodic) k-ary words with length less or equal to n (n,k >= 1).
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2
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1, 2, 1, 3, 4, 1, 4, 9, 10, 1, 5, 16, 33, 22, 1, 6, 25, 76, 105, 52, 1, 7, 36, 145, 316, 345, 106, 1, 8, 49, 246, 745, 1336, 1041, 232, 1, 9, 64, 385, 1506, 3865, 5356, 3225, 472, 1, 10, 81, 568, 2737, 9276, 19345, 21736, 9705, 976, 1, 11, 100, 801, 4600, 19537, 55686
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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 -k+k^3+k^4.
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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 * mu(j/d).
T(n,k) = Sum_{1<=j<=n} A143324(j,k).
T(n,k) = A143327(n,k) * k.
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EXAMPLE
| T(2,3) = 9, because there are 9 primitive words of length less or equal to 2 over 3-letter alphabet {a,b,c}: 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 ...
1, 4, 9, 16, 25 ...
1, 10, 33, 76, 145 ...
1, 22, 105, 316, 745 ...
1, 52, 345, 1336, 3865 ...
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MAPLE
| with (numtheory): f0:= proc (n) option remember; unapply (k^n-add(f0(d)(k), d=divisors(n) minus {n}), k) end; g0:= proc(n) option remember; unapply (add (f0(j)(x), j=1..n), x) end; T:= (n, k)-> g0(n)(k); seq (seq (T(i, d-i), i=1..d-1), d=2..13);
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CROSSREFS
| Column 1: A000012. Rows 1-3: A000027, A000290, A081437 and A085490. See also A143324, A143327, A134541, A008683.
Sequence in context: A051128 A137614 A204213 * A186686 A053122 A078812
Adjacent sequences: A143323 A143324 A143325 * A143327 A143328 A143329
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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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