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A320295 Number of series-reduced rooted trees whose leaves are non-singleton integer partitions whose multiset union is an integer partition of n.

%I A320295 #8 Oct 25 2018 22:22:13
%S A320295 0,1,2,5,8,19,34,80,165,394,892,2192,5232,13057,32271,81568,205748,
%T A320295 525735,1344828,3467415,8960849,23280323,60639680,158559047,415631368,
%U A320295 1092734050,2879420753,7605713020,20130266302,53386744298,141836904569,377479973474,1006189769886
%N A320295 Number of series-reduced rooted trees whose leaves are non-singleton integer partitions whose multiset union is an integer partition of n.
%C A320295 Also phylogenetic trees with no singleton leaves on integer partitions of n.
%H A320295 Andrew Howroyd, <a href="/A320295/b320295.txt">Table of n, a(n) for n = 1..500</a>
%e A320295 The a(2) = 1 through a(6) = 19 trees:
%e A320295   (11)  (21)   (22)        (32)         (33)
%e A320295         (111)  (31)        (41)         (42)
%e A320295                (211)       (221)        (51)
%e A320295                (1111)      (311)        (222)
%e A320295                ((11)(11))  (2111)       (321)
%e A320295                            (11111)      (411)
%e A320295                            ((11)(12))   (2211)
%e A320295                            ((11)(111))  (3111)
%e A320295                                         (21111)
%e A320295                                         (111111)
%e A320295                                         ((11)(13))
%e A320295                                         ((11)(22))
%e A320295                                         ((12)(12))
%e A320295                                         ((11)(112))
%e A320295                                         ((12)(111))
%e A320295                                         ((11)(1111))
%e A320295                                         ((111)(111))
%e A320295                                         ((11)(11)(11))
%e A320295                                         ((11)((11)(11)))
%t A320295 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A320295 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A320295 pgtm[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[pgtm/@p]],{p,Select[mps[m],Length[#]>1&]}],m];
%t A320295 Table[Sum[Length[Select[pgtm[m],FreeQ[#,{_}]&]],{m,IntegerPartitions[n]}],{n,14}]
%o A320295 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A320295 seq(n)={my(p=1/prod(k=1, n, 1 - x^k + O(x*x^n)), v=vector(n)); for(n=1, n, v[n]=polcoef(p, n) - 1 + EulerT(v[1..n])[n]); v} \\ _Andrew Howroyd_, Oct 25 2018
%Y A320295 Cf. A000311, A000669, A005804, A141268, A302545, A304966, A319312, A320289, A320294.
%K A320295 nonn
%O A320295 1,3
%A A320295 _Gus Wiseman_, Oct 09 2018
%E A320295 Terms a(12) and beyond from _Andrew Howroyd_, Oct 25 2018