A143327 Table T(n,k) by antidiagonals. T(n,k) is the number of primitive (=aperiodic) k-ary words (n,k >= 1) with length less than or equal to n which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

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

Offset

1,5

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.

Links

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Index entries for sequences related to Lyndon words

Formula

T(n,k) = Sum_{j=1..n} Sum_{d|j} k^(d-1) * mu(j/d).

T(n,k) = Sum_{j=1..n} A143325(j,k).

T(n,k) = A143326(n,k) / k.

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,      1,       1, ...
  1,   2,    3,     4,     5,      6,      7,       8, ...
  1,   5,   11,    19,    29,     41,     55,      71, ...
  1,  11,   35,    79,   149,    251,    391,     575, ...
  1,  26,  115,   334,   773,   1546,   2791,    4670, ...
  1,  53,  347,  1339,  3869,   9281,  19543,   37367, ...
  1, 116, 1075,  5434, 19493,  55936, 137191,  299510, ...
  1, 236, 3235, 21754, 97493, 335656, 960391, 2396150, ...

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(n, 1+d-n), n=1..d), d=1..12);

Mathematica

t[n_, k_] := Sum[k^(d-1)*MoebiusMu[j/d], {j, 1, n}, {d, Divisors[j]}]; Table[t[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 13 2013 *)

Crossrefs

Columns k=1-10 give: A000012, A085945, A320087, A320088, A320089, A320090, A320091, A320092, A320093, A320094.

Rows n=1-4 give: A000012, A000027, A028387, A003777.

Main diagonal gives A320095.

Cf. A143325, A143326, A134541, A008683.

Sequence in context: A049513 A121207 A097084 * A094954 A083064 A204057

Adjacent sequences: A143324 A143325 A143326 * A143328 A143329 A143330

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 07 2008

Status

approved