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A325550 Number of necklace compositions of n with distinct multiplicities.

%I A325550 #11 Aug 31 2019 22:26:15
%S A325550 1,2,2,4,5,7,11,16,18,41,86,118,273,465,731,1432,2791,4063,8429,14761,
%T A325550 29465,58654,123799,227419,453229,861909,1697645,3192807,6315007,
%U A325550 11718879,22795272,42965245,83615516,156215020,306561088,587300503,1140650287,2203107028
%N A325550 Number of necklace compositions of n with distinct multiplicities.
%C A325550 A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations.
%H A325550 Andrew Howroyd, <a href="/A325550/b325550.txt">Table of n, a(n) for n = 1..100</a>
%F A325550 a(n) = Sum_{d|n} phi(d)*(Sum_{k=1..n/d} A242887(n/d, k)/k)/d. - _Andrew Howroyd_, Aug 31 2019
%e A325550 The a(1) = 1 through a(8) = 16 necklace compositions:
%e A325550   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A325550        (11)  (111)  (22)    (113)    (33)      (115)      (44)
%e A325550                     (112)   (122)    (114)     (133)      (116)
%e A325550                     (1111)  (1112)   (222)     (223)      (224)
%e A325550                             (11111)  (1113)    (1114)     (233)
%e A325550                                      (11112)   (1222)     (1115)
%e A325550                                      (111111)  (11113)    (2222)
%e A325550                                                (11122)    (11114)
%e A325550                                                (11212)    (11222)
%e A325550                                                (111112)   (12122)
%e A325550                                                (1111111)  (111113)
%e A325550                                                           (111122)
%e A325550                                                           (111212)
%e A325550                                                           (112112)
%e A325550                                                           (1111112)
%e A325550                                                           (11111111)
%t A325550 neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
%t A325550 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&UnsameQ@@Length/@Split[Sort[#]]&]],{n,15}]
%o A325550 (PARI)
%o A325550 b(n)={((r,k,b,w)->if(!k||!r, if(r,0,(w-1)!), sum(m=0, r\k, if(!m || !bittest(b,m), self()(r-k*m, k-1, bitor(b,1<<m), w+m)/m!))))(n,n,1,0)}
%o A325550 a(n)={sumdiv(n, d, eulerphi(d)*b(n/d)/d)} \\ _Andrew Howroyd_, Aug 31 2019
%Y A325550 Cf. A000079, A000740, A008965, A059966, A098504, A098859, A242882, A242887, A325549, A325554.
%K A325550 nonn
%O A325550 1,2
%A A325550 _Gus Wiseman_, May 10 2019
%E A325550 Terms a(26) and beyond from _Andrew Howroyd_, Aug 31 2019