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A299027 Number of compositions of n whose standard factorization into Lyndon words has all distinct weakly increasing factors. 4
1, 1, 3, 5, 11, 20, 38, 69, 125, 225, 400, 708, 1244, 2176, 3779, 6532, 11229, 19223, 32745, 55555, 93875, 158025, 265038, 443009, 738026, 1225649, 2029305, 3350167, 5515384, 9055678, 14830076, 24226115, 39480306, 64190026, 104130753, 168556588, 272268482 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1000

FORMULA

Weigh transform of A167934.

EXAMPLE

The a(5) = 11 compositions:

      (5) = (5)

     (41) = (4)*(1)

     (14) = (14)

     (32) = (3)*(2)

     (23) = (23)

    (131) = (13)*(1)

    (113) = (113)

    (212) = (2)*(12)

    (122) = (122)

   (1121) = (112)*(1)

   (1112) = (1112)

Not included:

    (311) = (3)*(1)*(1)

    (221) = (2)*(2)*(1)

   (2111) = (2)*(1)*(1)*(1)

   (1211) = (12)*(1)*(1)

  (11111) = (1)*(1)*(1)*(1)*(1)

MATHEMATICA

nn=50;

ser=Product[(1+x^n)^(PartitionsP[n]-DivisorSigma[0, n]+1), {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(vector(n, n, numbpart(n) - numdiv(n) + 1))} \\ Andrew Howroyd, Dec 01 2018

CROSSREFS

Cf. A001045, A032153, A034691, A049311, A059966, A098407, A116540, A185700, A270995, A283877,  A292432, A293993, A296373, A299023, A299024, A299026.

Sequence in context: A058932 A118037 A094588 * A339006 A247353 A293948

Adjacent sequences:  A299024 A299025 A299026 * A299028 A299029 A299030

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 01 2018

STATUS

approved

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Last modified August 1 18:37 EDT 2021. Contains 346402 sequences. (Running on oeis4.)