cp's OEIS Frontend

A303551 Number of aperiodic multisets of compositions of total weight n.

%I A303551 #11 Sep 17 2018 03:17:13
%S A303551 1,2,6,15,41,95,243,567,1366,3189,7532,17428,40590,93465,215331,
%T A303551 493150,1127978,2569049,5841442,13240351,29953601,67596500,152258270,
%U A303551 342235866,767895382,1719813753,3845442485,8584197657,19133459138,42583565928,94641591888
%N A303551 Number of aperiodic multisets of compositions of total weight n.
%C A303551 A multiset is aperiodic if its multiplicities are relatively prime.
%H A303551 Andrew Howroyd, <a href="/A303551/b303551.txt">Table of n, a(n) for n = 1..1000</a>
%F A303551 a(n) = Sum_{d|n} mu(d) * A034691(n/d).
%e A303551 The a(4) = 15 aperiodic multisets of compositions are:
%e A303551 {4}, {31}, {22}, {211}, {13}, {121}, {112}, {1111},
%e A303551 {1,3}, {1,21}, {1,12}, {1,111}, {2,11},
%e A303551 {1,1,2}, {1,1,11}.
%e A303551 Missing from this list are {1,1,1,1}, {2,2}, and {11,11}.
%p A303551 with(numtheory):
%p A303551 b:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A303551       d*2^(d-1), d=divisors(j))*b(n-j), j=1..n)/n)
%p A303551     end:
%p A303551 a:= n-> add(mobius(d)*b(n/d), d=divisors(n)):
%p A303551 seq(a(n), n=1..35);  # _Alois P. Heinz_, Apr 26 2018
%t A303551 nn=20;
%t A303551 ser=Product[1/(1-x^n)^2^(n-1),{n,nn}]
%t A303551 Table[Sum[MoebiusMu[d]*SeriesCoefficient[ser,{x,0,n/d}],{d,Divisors[n]}],{n,1,nn}]
%o A303551 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A303551 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 A303551 Cf. A000740, A000837, A007716, A007916, A034691, A100953, A255906, A269134, A301700, A303386, A303431, A303546, A303552.
%K A303551 nonn
%O A303551 1,2
%A A303551 _Gus Wiseman_, Apr 26 2018