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A356933 Number of multisets of multisets, each of odd size, whose multiset union is a size-n multiset covering an initial interval.

%I A356933 #11 Jan 01 2023 17:58:57
%S A356933 1,1,2,8,28,108,524,2608,14176,86576,550672,3782496,27843880,
%T A356933 214071392,1751823600,15041687664,134843207240,1269731540864,
%U A356933 12427331494304,126619822952928,1341762163389920,14712726577081248,167209881188545344,1963715680476759040,23794190474350155856
%N A356933 Number of multisets of multisets, each of odd size, whose multiset union is a size-n multiset covering an initial interval.
%H A356933 Andrew Howroyd, <a href="/A356933/b356933.txt">Table of n, a(n) for n = 0..500</a>
%H A356933 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vR-C_picqWlu0KOguRGWaPjhS2HY7m43aGXGDcolDh4Qtyy-pu2lkq5mbHAbiMSyQoiIESG2mCGtc2j/pub">Counting and ranking classes of multiset partitions related to gapless multisets</a>
%e A356933 The a(4) = 28 multiset partitions:
%e A356933   {1}{111}      {1}{112}      {1}{123}      {1}{234}
%e A356933   {1}{1}{1}{1}  {1}{122}      {1}{223}      {2}{134}
%e A356933                 {1}{222}      {1}{233}      {3}{124}
%e A356933                 {2}{111}      {2}{113}      {4}{123}
%e A356933                 {2}{112}      {2}{123}      {1}{2}{3}{4}
%e A356933                 {2}{122}      {2}{133}
%e A356933                 {1}{1}{1}{2}  {3}{112}
%e A356933                 {1}{1}{2}{2}  {3}{122}
%e A356933                 {1}{2}{2}{2}  {3}{123}
%e A356933                               {1}{1}{2}{3}
%e A356933                               {1}{2}{2}{3}
%e A356933                               {1}{2}{3}{3}
%t A356933 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A356933 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A356933 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t A356933 Table[Length[Select[Join@@mps/@allnorm[n],OddQ[Times@@Length/@#]&]],{n,0,5}]
%o A356933 (PARI)
%o A356933 EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A356933 R(n,k) = {EulerT(vector(n, j, if(j%2 == 1, binomial(j+k-1, j))))}
%o A356933 seq(n) = {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ _Andrew Howroyd_, Jan 01 2023
%Y A356933 A000041 counts integer partitions, strict A000009.
%Y A356933 A000670 counts patterns, ranked by A333217, necklace A019536.
%Y A356933 A011782 counts multisets covering an initial interval.
%Y A356933 Cf. A055887, A063834, A072233, A270995, A304969, A349050, A349055.
%Y A356933 Odd-size multisets are counted by A000302, A027193, A058695, ranked by A026424.
%Y A356933 Other conditions: A034691, A116540, A255906, A356937, A356942.
%Y A356933 Other types: A050330, A356932, A356934, A356935.
%K A356933 nonn
%O A356933 0,3
%A A356933 _Gus Wiseman_, Sep 08 2022
%E A356933 Terms a(9) and beyond from _Andrew Howroyd_, Jan 01 2023