A299027 Number of compositions of n whose standard factorization into Lyndon words has all distinct weakly increasing factors.

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

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