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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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25
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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, 65280, 19656, 990, 0
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OFFSET
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1,2
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COMMENTS
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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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Alois P. Heinz, Antidiagonals n = 1..141, flattened
C. J. Smyth, A coloring proof of a generalisation of Fermat's little theorem, Amer. Math. Monthly 93, No. 6 (1986), pp. 469-471.
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.
So Sum_{d|n} k^d * mu(n/d) == 0 (mod n), this is a generalization of Fermat's little theorem k^p - k == 0 (mod p) for primes p to an arbitrary modulus n (see the Smyth link). - Franz Vrabec, Feb 09 2021
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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(n, 1+d-n), n=1..d), d=1..12);
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MATHEMATICA
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f0[n_] := f0[n] = Function [k, k^n - Sum[f0[d][k], {d, Complement[Divisors[n], {n}]}]]; t[n_, k_] := f0[n][k]; Table[Table[t[n, 1 + d - n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)
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CROSSREFS
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Columns k=1-10 give: A000007(n-1), A027375, A054718, A054719, A054720, A054721, A218124, A218125, A218126, A218127.
Rows n=1-10 give: A000027, A002378(k-1), A007531(k+1), A047928(k+1), A061167, A218130, A133499, A218131, A218132, A218133.
Main diagonal gives A252764.
Cf. A074650, A143325, A008683, A054525.
Sequence in context: A115241 A154559 A269133 * A287416 A097418 A154752
Adjacent sequences: A143321 A143322 A143323 * 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, Aug 07 2008
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STATUS
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approved
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