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A299024 Number of compositions of n whose standard factorization into Lyndon words has distinct strict compositions as factors. 4
1, 1, 3, 4, 7, 13, 21, 34, 58, 98, 158, 258, 421, 676, 1108, 1777, 2836, 4544, 7220, 11443, 18215, 28729, 45203, 71139, 111518, 174402, 272367, 424892, 660563, 1025717, 1590448, 2460346, 3800816, 5862640, 9026963, 13885425, 21321663, 32695098, 50073855 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..500

FORMULA

Weigh transform of A032153.

EXAMPLE

The a(5) = 7 compositions:

      (5) = (5)

     (41) = (4)*(1)

     (14) = (14)

     (32) = (3)*(2)

     (23) = (23)

    (131) = (13)*(1)

    (212) = (2)*(12)

Not included:

    (311) = (3)*(1)*(1)

    (113) = (113)

    (221) = (2)*(2)*(1)

    (122) = (122)

   (2111) = (2)*(1)*(1)*(1)

   (1211) = (12)*(1)*(1)

   (1121) = (112)*(1)

   (1112) = (1112)

  (11111) = (1)*(1)*(1)*(1)*(1)

MATHEMATICA

nn=50;

ser=Product[(1+x^n)^Total[(Length[#]-1)!&/@Select[IntegerPartitions[n], UnsameQ@@#&]], {n, nn}];

Table[SeriesCoefficient[ser, {x, 0, n}], {n, nn}]

PROG

(PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}

seq(N)={WeighT(Vec(sum(n=1, N-1, (n-1)!*x^(n*(n+1)/2)/prod(k=1, n, 1-x^k + O(x^N)))))} \\ Andrew Howroyd, Dec 01 2018

CROSSREFS

Cf. A001045, A032020, A032153, A034691, A049311, A050342, A059966, A089259, A098407, A116540, A185700, A270995, A283877,  A292432, A296373, A299023, A299026, A299027.

Sequence in context: A189994 A125118 A310008 * A116201 A280224 A282718

Adjacent sequences:  A299021 A299022 A299023 * A299025 A299026 A299027

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 31 2018

STATUS

approved

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Last modified July 4 11:10 EDT 2020. Contains 335447 sequences. (Running on oeis4.)