A299026 Number of compositions of n whose standard factorization into Lyndon words has all weakly increasing factors.

1, 2, 4, 8, 16, 31, 59, 111, 205, 378, 685, 1238, 2213, 3940, 6955, 12221, 21333, 37074, 64073, 110267, 188877, 322244, 547522, 926903, 1563370, 2628008, 4402927, 7353656, 12244434, 20329271, 33657560, 55574996, 91525882, 150356718, 246403694, 402861907

Offset

1,2

Links

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

Formula

Euler transform of A167934.

Example

The 2^6 - a(7) = 5 compositions of 7 whose Lyndon prime factors are not all weakly increasing: (11212), (1132), (1213), (1321), (142).

Mathematica

nn=50;
ser=Product[1/(1-x^n)^(PartitionsP[n]-DivisorSigma[0, n]+1), {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(vector(n, n, numbpart(n) - numdiv(n) + 1))} \\ Andrew Howroyd, Dec 01 2018

Crossrefs

Cf. A001045, A032153, A034691, A049311, A059966, A098407, A167934, A185700, A270995, A296373, A299023, A299024, A299027.

Sequence in context: A000128 A106399 A334636 * A007800 A362063 A309982

Adjacent sequences: A299023 A299024 A299025 * A299027 A299028 A299029

Keyword

nonn

Author

Gus Wiseman, Feb 01 2018

Status

approved