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A330655 Number of balanced reduced multisystems of weight n whose atoms cover an initial interval of positive integers.

%I A330655 #8 Dec 31 2019 06:49:41
%S A330655 1,1,2,12,138,2652,78106,3256404,182463296,13219108288,1202200963522,
%T A330655 134070195402644,17989233145940910,2858771262108762492,
%U A330655 530972857546678902490,113965195745030648131036,27991663753030583516229824,7800669355870672032684666900,2448021231611414334414904013956
%N A330655 Number of balanced reduced multisystems of weight n whose atoms cover an initial interval of positive integers.
%C A330655 A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.
%H A330655 Andrew Howroyd, <a href="/A330655/b330655.txt">Table of n, a(n) for n = 0..200</a>
%e A330655 The a(0) = 1 through a(3) = 12 multisystems:
%e A330655   {}  {1}  {1,1}  {1,1,1}
%e A330655            {1,2}  {1,1,2}
%e A330655                   {1,2,2}
%e A330655                   {1,2,3}
%e A330655                   {{1},{1,1}}
%e A330655                   {{1},{1,2}}
%e A330655                   {{1},{2,2}}
%e A330655                   {{1},{2,3}}
%e A330655                   {{2},{1,1}}
%e A330655                   {{2},{1,2}}
%e A330655                   {{2},{1,3}}
%e A330655                   {{3},{1,2}}
%t A330655 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t A330655 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A330655 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A330655 totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1<Length[#]<Length[m]&]}],m];
%t A330655 Table[Sum[Length[totm[m]],{m,allnorm[n]}],{n,0,5}]
%o A330655 (PARI)
%o A330655 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A330655 R(n,k)={my(v=vector(n), u=vector(n)); v[1]=k; for(n=1, #v, u += v*sum(j=n, #v, (-1)^(j-n)*binomial(j-1,n-1)); v=EulerT(v)); u}
%o A330655 seq(n)={concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k))))} \\ _Andrew Howroyd_, Dec 30 2019
%Y A330655 Row sums of A330776.
%Y A330655 The unlabeled version is A330474.
%Y A330655 The strongly normal case is A330475.
%Y A330655 The tree version is A330654.
%Y A330655 The maximum-depth case is A330676.
%Y A330655 The case where the atoms are all different is A005121.
%Y A330655 The case where the atoms are all equal is A318813.
%Y A330655 Multiset partitions of normal multisets are A255906.
%Y A330655 Series-reduced rooted trees with normal leaves are A316651.
%Y A330655 Cf. A000311, A000669, A001678, A213427, A318812, A330675, A330679.
%K A330655 nonn
%O A330655 0,3
%A A330655 _Gus Wiseman_, Dec 27 2019
%E A330655 Terms a(7) and beyond from _Andrew Howroyd_, Dec 30 2019