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%I #8 Jan 16 2023 21:54:01
%S 1,1,3,8,28,110,519,2749,16317,106425,755425,5781956,47384170,
%T 413331955,3818838624,37213866876,381108145231,4088785729738,
%U 45829237977692,535340785268513,6502943193997922,81984445333355812,1070848034863526547,14467833457108560375,201894571410270034773
%N Number of connected multiset partitions of normal multisets of size n.
%C A multiset is normal if it spans an initial interval of positive integers.
%H Andrew Howroyd, <a href="/A317077/b317077.txt">Table of n, a(n) for n = 0..500</a>
%e The a(3) = 8 connected multiset partitions are (111), (1)(11), (1)(1)(1), (122), (2)(12), (112), (1)(12), (123).
%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 csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],multijoin@@s[[c[[1]]]]]]]]];
%t allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
%t Length/@Table[Join@@Table[Select[mps[m],Length[csm[#]]==1&],{m,allnorm[n]}],{n,8}]
%o (PARI)
%o EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o Connected(v)={my(u=vector(#v));for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1,k)*v[k]*u[n-k]));u}
%o seq(n)={my(u=vector(n, k, x*Ser(EulerT(vector(n,i,binomial(i+k-1,i)))))); Vec(1+vecsum(Connected(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k,i)*u[i])))))} \\ _Andrew Howroyd_, Jan 16 2023
%Y Cf. A007716, A007718, A048143, A293994, A303837, A303838, A304716, A305078.
%Y Cf. A317073, A317075, A317078, A317079, A317080.
%K nonn
%O 0,3
%A _Gus Wiseman_, Jul 20 2018
%E Terms a(9) and beyond from _Andrew Howroyd_, Jan 16 2023
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