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%I #14 Dec 30 2020 14:34:36
%S 1,1,2,8,70,1012,21944,665708,26917492,1399033348,90878863352,
%T 7214384973908,687197223963640,77354805301801012,10158257981179981304,
%U 1539156284259756811748,266517060496258245459352,52301515332984084095078308,11546416513975694879642736152
%N Number of balanced reduced multisystems of weight n and maximum depth whose atoms cover an initial interval of positive integers.
%C 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.
%C A finite multiset is normal if it covers an initial interval of positive integers.
%H Andrew Howroyd, <a href="/A330676/b330676.txt">Table of n, a(n) for n = 0..200</a>
%e The a(0) = 1 through a(3) = 8 multisystems:
%e {} {1} {1,1} {{1},{1,1}}
%e {1,2} {{1},{1,2}}
%e {{1},{2,2}}
%e {{1},{2,3}}
%e {{2},{1,1}}
%e {{2},{1,2}}
%e {{2},{1,3}}
%e {{3},{1,2}}
%t allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-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 totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1<Length[#]<Length[m]&]}],m];
%t Table[Sum[Length[Select[totm[m],Depth[#]==If[n<=1,2,n]&]],{m,allnorm[n]}],{n,5}]
%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o R(n, k)={my(v=vector(n), u=vector(n)); v[1]=k; for(n=1, #v, for(i=n, #v, u[i] += v[i]*(-1)^(i-n)*binomial(i-1, n-1)); v=EulerT(v)); u}
%o 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 2020
%Y Row sums of A330778.
%Y The case with all atoms equal is A000111.
%Y The case with all atoms different is A006472.
%Y The version allowing all depths is A330655.
%Y The unlabeled version is A330663.
%Y The version where the atoms are the prime indices of n is A330665.
%Y The strongly normal version is A330675.
%Y The version where the degrees are the prime indices of n is A330728.
%Y Multiset partitions of normal multisets are A255906.
%Y Series-reduced rooted trees with normal leaves are A316651.
%Y Cf. A000669, A001055, A005121, A005804, A318812, A330469, A330474, A330654, A330664, A330677, A330679.
%K nonn
%O 0,3
%A _Gus Wiseman_, Dec 30 2019
%E Terms a(8) and beyond from _Andrew Howroyd_, Dec 30 2019
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