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A296373 Triangle T(n,k) = number of compositions of n whose factorization into Lyndon words (aperiodic necklaces) is of length k. 14
1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 6, 5, 3, 1, 1, 9, 12, 6, 3, 1, 1, 18, 21, 14, 6, 3, 1, 1, 30, 45, 27, 15, 6, 3, 1, 1, 56, 84, 61, 29, 15, 6, 3, 1, 1, 99, 170, 120, 67, 30, 15, 6, 3, 1, 1, 186, 323, 254, 136, 69, 30, 15, 6, 3, 1, 1, 335, 640, 510, 295, 142, 70, 30, 15, 6, 3, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Wikipedia, Lyndon word: Standard factorization

FORMULA

First column is A059966.

EXAMPLE

Triangle begins:

    1;

    1,   1;

    2,   1,   1;

    3,   3,   1,   1;

    6,   5,   3,   1,   1;

    9,  12,   6,   3,   1,   1;

   18,  21,  14,   6,   3,   1,   1;

   30,  45,  27,  15,   6,   3,   1,   1;

   56,  84,  61,  29,  15,   6,   3,   1,   1;

   99, 170, 120,  67,  30,  15,   6,   3,   1,   1;

  186, 323, 254, 136,  69,  30,  15,   6,   3,   1,   1;

  335, 640, 510, 295, 142,  70,  30,  15,   6,   3,   1,   1;

MATHEMATICA

neckQ[q_]:=Array[OrderedQ[{RotateRight[q, #], q}]&, Length[q]-1, 1, And];

aperQ[q_]:=UnsameQ@@Table[RotateRight[q, k], {k, Length[q]}];

qit[q_]:=If[#===Length[q], {q}, Prepend[qit[Drop[q, #]], Take[q, #]]]&[Max@@Select[Range[Length[q]], neckQ[Take[q, #]]&&aperQ[Take[q, #]]&]];

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Length[qit[#]]===k&]], {n, 12}, {k, n}]

PROG

(PARI) EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i ))-1)}

A(n)=[Vecrev(p/y) | p<-EulerMT(y*vector(n, n, sumdiv(n, d, moebius(n/d) * (2^d-1))/n))]

{ my(T=A(12)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Dec 01 2018

CROSSREFS

Cf. A000740, A001045, A008965, A019536, A059966, A060223, A185700, A228369, A232472, A277427, A281013, A296302, A296372.

Sequence in context: A304942 A090011 A061554 * A088326 A216956 A285522

Adjacent sequences:  A296370 A296371 A296372 * A296374 A296375 A296376

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Dec 11 2017

STATUS

approved

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Last modified February 16 21:59 EST 2019. Contains 320200 sequences. (Running on oeis4.)