A320295 - OEIS

%I #8 Oct 25 2018 22:22:13

%S 0,1,2,5,8,19,34,80,165,394,892,2192,5232,13057,32271,81568,205748,

%T 525735,1344828,3467415,8960849,23280323,60639680,158559047,415631368,

%U 1092734050,2879420753,7605713020,20130266302,53386744298,141836904569,377479973474,1006189769886

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

%C Also phylogenetic trees with no singleton leaves on integer partitions of n.

%H Andrew Howroyd, <a href="/A320295/b320295.txt">Table of n, a(n) for n = 1..500</a>

%e The a(2) = 1 through a(6) = 19 trees:

%e   (11)  (21)   (22)        (32)         (33)

%e         (111)  (31)        (41)         (42)

%e                (211)       (221)        (51)

%e                (1111)      (311)        (222)

%e                ((11)(11))  (2111)       (321)

%e                            (11111)      (411)

%e                            ((11)(12))   (2211)

%e                            ((11)(111))  (3111)

%e                                         (21111)

%e                                         (111111)

%e                                         ((11)(13))

%e                                         ((11)(22))

%e                                         ((12)(12))

%e                                         ((11)(112))

%e                                         ((12)(111))

%e                                         ((11)(1111))

%e                                         ((111)(111))

%e                                         ((11)(11)(11))

%e                                         ((11)((11)(11)))

%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 pgtm[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[pgtm/@p]],{p,Select[mps[m],Length[#]>1&]}],m];

%t Table[Sum[Length[Select[pgtm[m],FreeQ[#,{_}]&]],{m,IntegerPartitions[n]}],{n,14}]

%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}

%o 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 Cf. A000311, A000669, A005804, A141268, A302545, A304966, A319312, A320289, A320294.

%K nonn

%O 1,3

%A _Gus Wiseman_, Oct 09 2018

%E Terms a(12) and beyond from _Andrew Howroyd_, Oct 25 2018