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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

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) a(n)={sum(k=1, n, sumdiv(k, d, moebius(k/d)*sum(i=1, d, binomial(d-1, i-1)*binomial(n-k+i, i))))} \\ Andrew Howroyd, Sep 18 2018

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 April 25 16:08 EDT 2019. Contains 322461 sequences. (Running on oeis4.)