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%I A319001 #9 Jan 16 2023 22:51:34
%S A319001 1,1,3,7,18,42,105,248,606,1450,3507,8415,20305,48785,117502,282574,
%T A319001 680137,1636005,3936841,9470776,22787529,54822530,131901491,317336519,
%U A319001 763489051,1836862947,4419324581,10632404189,25580507505,61543948594,148068421107
%N A319001 Number of ordered multiset partitions of integer partitions of n where the sequence of GCDs of the partitions is weakly increasing.
%C A319001 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 A319001 Andrew Howroyd, <a href="/A319001/b319001.txt">Table of n, a(n) for n = 0..1000</a>
%e A319001 The a(4) = 18 ordered multiset partitions:
%e A319001   {{4}}   {{1,3}}    {{2,2}}     {{1,1,2}}       {{1,1,1,1}}
%e A319001          {{1},{3}}  {{2},{2}}   {{1},{1,2}}     {{1},{1,1,1}}
%e A319001                                 {{1,2},{1}}     {{1,1,1},{1}}
%e A319001                                 {{1,1},{2}}     {{1,1},{1,1}}
%e A319001                                {{1},{1},{2}}   {{1},{1},{1,1}}
%e A319001                                                {{1},{1,1},{1}}
%e A319001                                                {{1,1},{1},{1}}
%e A319001                                               {{1},{1},{1},{1}}
%o A319001 (PARI) \\ here B(n) is A000837 as vector.
%o A319001 B(n) = {dirmul(vector(n, k, moebius(k)), vector(n, k, numbpart(k)))}
%o A319001 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 A319001 Cf. A000837, A007716, A055887, A063834, A255397, A269134, A276024, A289508, A316222, A317545, A317546, A319002, A319003.
%K A319001 nonn
%O A319001 0,3
%A A319001 _Gus Wiseman_, Sep 07 2018
%E A319001 a(0)=1 prepended and terms a(11) and beyond from _Andrew Howroyd_, Jan 16 2023

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