A303551 - OEIS

%I #11 Sep 17 2018 03:17:13

%S 1,2,6,15,41,95,243,567,1366,3189,7532,17428,40590,93465,215331,

%T 493150,1127978,2569049,5841442,13240351,29953601,67596500,152258270,

%U 342235866,767895382,1719813753,3845442485,8584197657,19133459138,42583565928,94641591888

%N Number of aperiodic multisets of compositions of total weight n.

%C A multiset is aperiodic if its multiplicities are relatively prime.

%H Andrew Howroyd, <a href="/A303551/b303551.txt">Table of n, a(n) for n = 1..1000</a>

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

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

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

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

%e {1,1,2}, {1,1,11}.

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

%p with(numtheory):

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

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

%p     end:

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

%p seq(a(n), n=1..35);  # _Alois P. Heinz_, Apr 26 2018

%t nn=20;

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

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

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

%o 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

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Apr 26 2018