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A074650
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Table T(n,k) by antidiagonals. Number of Lyndon words (aperiodic necklaces) with n beads of k colors.
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17
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1, 2, 0, 3, 1, 0, 4, 3, 2, 0, 5, 6, 8, 3, 0, 6, 10, 20, 18, 6, 0, 7, 15, 40, 60, 48, 9, 0, 8, 21, 70, 150, 204, 116, 18, 0, 9, 28, 112, 315, 624, 670, 312, 30, 0, 10, 36, 168, 588, 1554, 2580, 2340, 810, 56, 0, 11, 45, 240, 1008, 3360, 7735, 11160, 8160, 2184, 99, 0, 12
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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REFERENCES
| F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pg 97 (2.3.74)
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LINKS
| Index entries for sequences related to Lyndon words
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FORMULA
| T(n,k) = (1/n) * Sum ( mu(n/d)*k^d ), d|n
T(n,k) = (k^n - Sum_{d<n,d|n} d*T(d,k)) / n - Alois P. Heinz, Mar 28 2008
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EXAMPLE
| 1, 2, 3, 4, 5 ...
0, 1, 3, 6, 10 ...
0, 2, 8, 20, 40 ...
0, 3, 18, 60, 150 ...
0, 6, 48, 204, 624 ...
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MAPLE
| with (numtheory): T := proc (n, k) add(mobius(n/d)*k^d, d=divisors(n))/n end; seq (seq(T(i, d-i), i=1..d-1), d=2..12); # Alois P. Heinz, Mar 28 2008
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MATHEMATICA
| max = 12; t[n_, k_] := Total[ MoebiusMu[n/#]*k^# & /@ Divisors[n]]/n; Flatten[ Table[ t[n-k+1, k], {n, 1, max}, {k, n, 1, -1}]] (* From Jean-François Alcover, Oct 18 2011, after Maple *)
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PROG
| (PARI) T(n, k)=sumdiv(n, d, moebius(n/d)*k^d)/n \\ Charles R Greathouse IV, Oct 18 2011
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CROSSREFS
| Columns 2-12: A001037, A027376, A027377, A001692, A032164, A001693, A027380, A027381, A032165, A032166, A032167.
Rows 1-4: A000027, A000217(n-1), A007290(n+1), A006011.
Diagonal: A075147.
See also A102659.
Sequence in context: A003988 A185914 A144257 * A202064 A144955 A168020
Adjacent sequences: A074647 A074648 A074649 * A074651 A074652 A074653
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KEYWORD
| nonn,tabl
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AUTHOR
| Christian G. Bower (bowerc(AT)usa.net), Aug 28 2002
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