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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.
5
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
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 09 2018
EXTENSIONS
Terms a(12) and beyond from Andrew Howroyd, Oct 25 2018
STATUS
approved