%I #9 Jan 16 2023 22:51:34

%S 1,1,3,7,18,42,105,248,606,1450,3507,8415,20305,48785,117502,282574,

%T 680137,1636005,3936841,9470776,22787529,54822530,131901491,317336519,

%U 763489051,1836862947,4419324581,10632404189,25580507505,61543948594,148068421107

%N Number of ordered multiset partitions of integer partitions of n where the sequence of GCDs of the partitions is weakly increasing.

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

%H Andrew Howroyd, <a href="/A319001/b319001.txt">Table of n, a(n) for n = 0..1000</a>

%e The a(4) = 18 ordered multiset partitions:

%e {{4}} {{1,3}} {{2,2}} {{1,1,2}} {{1,1,1,1}}

%e {{1},{3}} {{2},{2}} {{1},{1,2}} {{1},{1,1,1}}

%e {{1,2},{1}} {{1,1,1},{1}}

%e {{1,1},{2}} {{1,1},{1,1}}

%e {{1},{1},{2}} {{1},{1},{1,1}}

%e {{1},{1,1},{1}}

%e {{1,1},{1},{1}}

%e {{1},{1},{1},{1}}

%o (PARI) \\ here B(n) is A000837 as vector.

%o B(n) = {dirmul(vector(n, k, moebius(k)), vector(n, k, numbpart(k)))}

%o seq(n) ={my(p=x*Ser(B(n))); Vec(1/prod(g=1, n, 1 - subst(p + O(x*x^(n\g)), x, x^g)))} \\ _Andrew Howroyd_, Jan 16 2023

%Y Cf. A000837, A007716, A055887, A063834, A255397, A269134, A276024, A289508, A316222, A317545, A317546, A319002, A319003.

%K nonn

%O 0,3

%A _Gus Wiseman_, Sep 07 2018

%E a(0)=1 prepended and terms a(11) and beyond from _Andrew Howroyd_, Jan 16 2023