A319003 Number of ordered multiset partitions of integer partitions of n where the sequence of LCMs of the blocks is weakly increasing.

1, 1, 3, 7, 17, 38, 87, 191, 420, 908, 1954, 4160, 8816, 18549, 38851, 80965, 168077, 347566, 716443, 1472344, 3017866, 6170789, 12590805, 25640050, 52122784, 105791068, 214413852, 434007488, 877480395, 1772235212, 3575967030, 7209301989, 14523006820

Offset

0,3

Comments

If we form a multiorder by treating integer partitions (a,...,z) as multiarrows LCM(a,...,z) <= {z,...,a}, then a(n) is the number of triangles of weight n.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..50

Example

The a(4) = 17 ordered multiset partitions:
  {{4}}   {{1,3}}    {{2,2}}     {{1,1,2}}      {{1,1,1,1}}
         {{1},{3}}  {{2},{2}}   {{1},{1,2}}    {{1},{1,1,1}}
                                {{1,1},{2}}    {{1,1,1},{1}}
                               {{1},{1},{2}}   {{1,1},{1,1}}
                                               {{1},{1},{1,1}}
                                               {{1},{1,1},{1}}
                                               {{1,1},{1},{1}}
                                              {{1},{1},{1},{1}}

Prog

(PARI) seq(n)={my(M=Map()); for(m=1, n, forpart(p=m, my(k=lcm(Vec(p)), z); mapput(M, k, if(mapisdefined(M, k, &z), z, 1 + O(x*x^n)) - x^m))); Vec(1/vecprod(Mat(M)[, 2]))} \\ Andrew Howroyd, Jan 16 2023

Crossrefs

Cf. A000837, A007716, A063834, A269134, A290103, A316222, A317545, A317546, A319001, A319004.

Sequence in context: A330457 A026396 A176502 * A141199 A368655 A003478

Adjacent sequences: A319000 A319001 A319002 * A319004 A319005 A319006

Keyword

nonn

Author

Gus Wiseman, Sep 07 2018

Extensions

a(0)=1 prepended and terms a(11) and beyond from Andrew Howroyd, Jan 16 2023

Status

approved