cp's OEIS Frontend

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

%I A319003 #10 Jan 16 2023 22:51:28
%S A319003 1,1,3,7,17,38,87,191,420,908,1954,4160,8816,18549,38851,80965,168077,
%T A319003 347566,716443,1472344,3017866,6170789,12590805,25640050,52122784,
%U A319003 105791068,214413852,434007488,877480395,1772235212,3575967030,7209301989,14523006820
%N A319003 Number of ordered multiset partitions of integer partitions of n where the sequence of LCMs of the blocks is weakly increasing.
%C A319003 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.
%H A319003 Andrew Howroyd, <a href="/A319003/b319003.txt">Table of n, a(n) for n = 0..50</a>
%e A319003 The a(4) = 17 ordered multiset partitions:
%e A319003   {{4}}   {{1,3}}    {{2,2}}     {{1,1,2}}      {{1,1,1,1}}
%e A319003          {{1},{3}}  {{2},{2}}   {{1},{1,2}}    {{1},{1,1,1}}
%e A319003                                 {{1,1},{2}}    {{1,1,1},{1}}
%e A319003                                {{1},{1},{2}}   {{1,1},{1,1}}
%e A319003                                                {{1},{1},{1,1}}
%e A319003                                                {{1},{1,1},{1}}
%e A319003                                                {{1,1},{1},{1}}
%e A319003                                               {{1},{1},{1},{1}}
%o A319003 (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
%Y A319003 Cf. A000837, A007716, A063834, A269134, A290103, A316222, A317545, A317546, A319001, A319004.
%K A319003 nonn
%O A319003 0,3
%A A319003 _Gus Wiseman_, Sep 07 2018
%E A319003 a(0)=1 prepended and terms a(11) and beyond from _Andrew Howroyd_, Jan 16 2023