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A303551 Number of aperiodic multisets of compositions of total weight n. 6
1, 2, 6, 15, 41, 95, 243, 567, 1366, 3189, 7532, 17428, 40590, 93465, 215331, 493150, 1127978, 2569049, 5841442, 13240351, 29953601, 67596500, 152258270, 342235866, 767895382, 1719813753, 3845442485, 8584197657, 19133459138, 42583565928, 94641591888 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A multiset is aperiodic if its multiplicities are relatively prime.

LINKS

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

FORMULA

a(n) = Sum_{d|n} mu(d) * A034691(n/d).

EXAMPLE

The a(4) = 15 aperiodic multisets of compositions are:

{4}, {31}, {22}, {211}, {13}, {121}, {112}, {1111},

{1,3}, {1,21}, {1,12}, {1,111}, {2,11},

{1,1,2}, {1,1,11}.

Missing from this list are {1,1,1,1}, {2,2}, and {11,11}.

MAPLE

with(numtheory):

b:= proc(n) option remember; `if`(n=0, 1, add(add(

      d*2^(d-1), d=divisors(j))*b(n-j), j=1..n)/n)

    end:

a:= n-> add(mobius(d)*b(n/d), d=divisors(n)):

seq(a(n), n=1..35);  # Alois P. Heinz, Apr 26 2018

MATHEMATICA

nn=20;

ser=Product[1/(1-x^n)^2^(n-1), {n, nn}]

Table[Sum[MoebiusMu[d]*SeriesCoefficient[ser, {x, 0, n/d}], {d, Divisors[n]}], {n, 1, nn}]

PROG

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

seq(n)={my(u=EulerT(vector(n, n, 2^(n-1)))); vector(n, n, sumdiv(n, d, moebius(d)*u[n/d]))} \\ Andrew Howroyd, Sep 15 2018

CROSSREFS

Cf. A000740, A000837, A007716, A007916, A034691, A100953, A255906, A269134, A301700, A303386, A303431, A303546, A303552.

Sequence in context: A061322 A004664 A074446 * A180666 A280788 A121328

Adjacent sequences:  A303548 A303549 A303550 * A303552 A303553 A303554

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 26 2018

STATUS

approved

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Last modified September 28 08:36 EDT 2021. Contains 347713 sequences. (Running on oeis4.)