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%I #11 Dec 31 2022 11:20:03
%S 0,1,2,4,7,14,28,54,106,208,399,757,1424,2642,4860,8851,15991,28673,
%T 51095,90454,159306,279067,486598,844514,1459625,2512227,4307409,
%U 7357347,12522304,21238683,35903463,60497684,101625958,170202949,284238857,473356564,786196353
%N Number of odd-length multiset partitions of integer partitions of n.
%H Andrew Howroyd, <a href="/A358837/b358837.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: ((1/Product_{k>=1} (1-x^k)^A000041(k)) - (1/Product_{k>=1} (1+x^k)^A000041(k))) / 2. - _Andrew Howroyd_, Dec 31 2022
%e The a(1) = 1 through a(5) = 14 multiset partitions:
%e {{1}} {{2}} {{3}} {{4}} {{5}}
%e {{1,1}} {{1,2}} {{1,3}} {{1,4}}
%e {{1,1,1}} {{2,2}} {{2,3}}
%e {{1},{1},{1}} {{1,1,2}} {{1,1,3}}
%e {{1,1,1,1}} {{1,2,2}}
%e {{1},{1},{2}} {{1,1,1,2}}
%e {{1},{1},{1,1}} {{1,1,1,1,1}}
%e {{1},{1},{3}}
%e {{1},{2},{2}}
%e {{1},{1},{1,2}}
%e {{1},{2},{1,1}}
%e {{1},{1},{1,1,1}}
%e {{1},{1,1},{1,1}}
%e {{1},{1},{1},{1},{1}}
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t Table[Length[Select[Join@@mps/@Reverse/@IntegerPartitions[n],OddQ[Length[#]]&]],{n,0,10}]
%o (PARI)
%o P(v,y) = {1/prod(k=1, #v, (1 - y*x^k + O(x*x^#v))^v[k])}
%o seq(n) = {my(v=vector(n, k, numbpart(k))); (Vec(P(v,1)) - Vec(P(v,-1)))/2} \\ _Andrew Howroyd_, Dec 31 2022
%Y The version for set partitions is A024429.
%Y These multiset partitions are ranked by A026424.
%Y The version for partitions is A027193.
%Y The version for twice-partitions is A358824.
%Y A001970 counts multiset partitions of integer partitions.
%Y A063834 counts twice-partitions, strict A296122.
%Y Cf. A000219, A141199, A336342, A358334, A358831.
%K nonn
%O 0,3
%A _Gus Wiseman_, Dec 05 2022
%E Terms a(11) and beyond from _Andrew Howroyd_, Dec 31 2022
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