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

0, 1, 2, 5, 8, 19, 34, 80, 165, 394, 892, 2192, 5232, 13057, 32271, 81568, 205748, 525735, 1344828, 3467415, 8960849, 23280323, 60639680, 158559047, 415631368, 1092734050, 2879420753, 7605713020, 20130266302, 53386744298, 141836904569, 377479973474, 1006189769886

Offset

1,3

Comments

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

Links

Andrew Howroyd, Table of n, a(n) for n = 1..500

Example

The a(2) = 1 through a(6) = 19 trees:
  (11)  (21)   (22)        (32)         (33)
        (111)  (31)        (41)         (42)
               (211)       (221)        (51)
               (1111)      (311)        (222)
               ((11)(11))  (2111)       (321)
                           (11111)      (411)
                           ((11)(12))   (2211)
                           ((11)(111))  (3111)
                                        (21111)
                                        (111111)
                                        ((11)(13))
                                        ((11)(22))
                                        ((12)(12))
                                        ((11)(112))
                                        ((12)(111))
                                        ((11)(1111))
                                        ((111)(111))
                                        ((11)(11)(11))
                                        ((11)((11)(11)))

Mathematica

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
pgtm[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[pgtm/@p]], {p, Select[mps[m], Length[#]>1&]}], m];
Table[Sum[Length[Select[pgtm[m], FreeQ[#, {_}]&]], {m, IntegerPartitions[n]}], {n, 14}]

Prog

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

Crossrefs

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

Sequence in context: A025078 A218937 A192780 * A277263 A055614 A243094

Adjacent sequences: A320292 A320293 A320294 * A320296 A320297 A320298

Keyword

nonn

Author

Gus Wiseman, Oct 09 2018

Extensions

Terms a(12) and beyond from Andrew Howroyd, Oct 25 2018

Status

approved