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%I #12 Feb 04 2021 16:30:43
%S 1,3,9,26,75,207,565,1518,4044,10703,28234,74277,195103,511902,
%T 1342147,3517239,9214412,24134528,63204417,165505811,433361425,
%U 1134664831,2970787794,7777975396,20363634815,53313819160,139579420528,365427311171,956707667616,2504704955181
%N Number of different aperiodic multisets that fit within some normal multiset of size n.
%C A multiset is normal if it spans an initial interval of positive integers. It is aperiodic if its multiplicities are relatively prime.
%H Andrew Howroyd, <a href="/A303976/b303976.txt">Table of n, a(n) for n = 1..500</a>
%F a(n) = Sum_{k=1..n} Sum_{d|k} mu(k/d) * Sum_{i=1..d} binomial(d-1, i-1)*binomial(n-k+i, i). - _Andrew Howroyd_, Sep 18 2018
%F G.f.: Sum_{d>=1} mu(d)*x^d/((1 - x - x^d*(2-x))*(1-x)). - _Andrew Howroyd_, Feb 04 2021
%e The a(4) = 26 aperiodic multisets:
%e (1), (2), (3), (4),
%e (12), (13), (14), (23), (24), (34),
%e (112), (113), (122), (123), (124), (133), (134), (223), (233), (234),
%e (1112), (1123), (1222), (1223), (1233), (1234).
%t allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
%t Table[Length[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]===1&]],{n,10}]
%o (PARI) seq(n)={Vec(sum(d=1, n, moebius(d)*x^d/(1 - x - x^d*(2-x)) + O(x*x^n))/(1-x))} \\ _Andrew Howroyd_, Feb 04 2021
%Y Row sums of A303974.
%Y Cf. A000740, A000837, A007916, A027941, A178472, A210554, A301700, A303431, A303546, A303551, A303945.
%K nonn
%O 1,2
%A _Gus Wiseman_, May 03 2018
%E Terms a(13) and beyond from _Andrew Howroyd_, Sep 18 2018