A299023 Number of compositions of n whose standard factorization into Lyndon words has all strict compositions as factors.

1, 2, 4, 7, 12, 23, 38, 66, 112, 193, 319, 539, 887, 1466, 2415, 3951, 6417, 10428, 16817, 27072, 43505, 69560, 110916, 176469, 279893, 442742, 698919, 1100898, 1729530, 2712134, 4244263, 6628174, 10332499, 16077835, 24972415, 38729239, 59958797, 92685287

Offset

1,2

Links

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

Formula

Euler transform of A032153.

Example

The a(5) = 12 compositions:
      (5) = (5)
     (41) = (4)*(1)
     (14) = (14)
     (32) = (3)*(2)
     (23) = (23)
    (311) = (3)*(1)*(1)
    (131) = (13)*(1)
    (221) = (2)*(2)*(1)
    (212) = (2)*(12)
   (2111) = (2)*(1)*(1)*(1)
   (1211) = (12)*(1)*(1)
  (11111) = (1)*(1)*(1)*(1)*(1)
Not included:
    (113) = (113)
    (122) = (122)
   (1121) = (112)*(1)
   (1112) = (1112)

Mathematica

nn=50;
ser=Product[1/(1-x^n)^Total[(Length[#]-1)!&/@Select[IntegerPartitions[n], UnsameQ@@#&]], {n, nn}];
Table[SeriesCoefficient[ser, {x, 0, n}], {n, nn}]

Prog

(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(N)={EulerT(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, A059966, A089259, A098407, A116540, A185700, A270995, A296373, A299024, A299026, A299027.

Sequence in context: A347017 A082548 A270995 * A007323 A099604 A026790

Adjacent sequences: A299020 A299021 A299022 * A299024 A299025 A299026

Keyword

nonn

Author

Gus Wiseman, Jan 31 2018

Status

approved