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 A303976 Number of different aperiodic multisets that fit within some normal multiset of size n. 4
 1, 3, 9, 26, 75, 207, 565, 1518, 4044, 10703, 28234, 74277, 195103, 511902, 1342147, 3517239, 9214412, 24134528, 63204417, 165505811, 433361425, 1134664831, 2970787794, 7777975396, 20363634815, 53313819160, 139579420528, 365427311171, 956707667616, 2504704955181 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS A multiset is normal if it spans an initial interval of positive integers. It is aperiodic if its multiplicities are relatively prime. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..500 FORMULA 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 G.f.: Sum_{d>=1} mu(d)*x^d/((1 - x - x^d*(2-x))*(1-x)). - Andrew Howroyd, Feb 04 2021 EXAMPLE The a(4) = 26 aperiodic multisets: (1), (2), (3), (4), (12), (13), (14), (23), (24), (34), (112), (113), (122), (123), (124), (133), (134), (223), (233), (234), (1112), (1123), (1222), (1223), (1233), (1234). MATHEMATICA allnorm[n_Integer]:=Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]; Table[Length[Select[Union@@Rest/@Subsets/@allnorm[n], GCD@@Length/@Split[#]===1&]], {n, 10}] PROG (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 CROSSREFS Row sums of A303974. Cf. A000740, A000837, A007916, A027941, A178472, A210554, A301700, A303431, A303546, A303551, A303945. Sequence in context: A276068 A171277 A289806 * A000243 A076264 A018919 Adjacent sequences:  A303973 A303974 A303975 * A303977 A303978 A303979 KEYWORD nonn AUTHOR Gus Wiseman, May 03 2018 EXTENSIONS Terms a(13) and beyond from Andrew Howroyd, Sep 18 2018 STATUS approved

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Last modified December 2 17:59 EST 2021. Contains 349445 sequences. (Running on oeis4.)