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Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.
25

%I #15 Jan 22 2021 16:49:37

%S 1,2,9,69,623,7793,110430,1906317,36833614,816101825,19925210834,

%T 541363267613,15997458049946,515769374925576,17905023985615254,

%U 669030297769291562,26689471638523499483,1134895275721374771655,51161002326406795249910,2440166138715867838359915

%N Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

%C A rooted tree is series-reduced if every non-leaf node has at least two branches.

%e The a(3) = 9 trees:

%e (1(11)), (111),

%e (1(12)), (2(11)), (112),

%e (1(23)), (2(13)), (3(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 gro[m_]:=If[Length[m]==1,m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];

%t Table[Sum[Length[gro[m]],{m,Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n]}],{n,4}]

%o (PARI) \\ See A339645 for combinatorial species functions.

%o cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sExp(x*Ser(v[1..n])), n )); x*Ser(v)}

%o StronglyNormalLabelingsSeq(cycleIndexSeries(15)) \\ _Andrew Howroyd_, Jan 04 2021

%Y Cf. A000081, A000311, A000669, A001678, A005804, A141268, A181821, A292504, A304660.

%Y Cf. A316651, A316653, A316654, A316655, A316656.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jul 09 2018

%E Terms a(10) and beyond from _Andrew Howroyd_, Jan 04 2021