cp's OEIS Frontend

A358837 Number of odd-length multiset partitions of integer partitions of n.

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