A303976 Number of different aperiodic multisets that fit within some normal multiset of size n.

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

Offset: 1

Author

Gus Wiseman, May 03 2018

Keywords

• nonn

Comments

A multiset is normal if it spans an initial interval of positive integers. It is aperiodic if its multiplicities are relatively prime.

Examples

			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).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..500

Crossrefs

Row sums of A303974.

Cf. A000740, A000837, A007916, A027941, A178472, A210554, A301700, A303431, A303546, A303551, A303945.

Programs

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}]

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

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

Extensions

Terms a(13) and beyond from Andrew Howroyd, Sep 18 2018