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A320178
Number of series-reduced rooted identity trees whose leaves are constant integer partitions whose multiset union is an integer partition of n.
5
1, 2, 4, 8, 19, 53, 151, 459, 1445, 4634, 15154, 50253, 168607, 571212, 1951588, 6715575, 23255444, 80978697, 283373024, 995995996, 3514614634, 12446666967, 44222390525, 157587392768, 563096832839, 2017121728223, 7242436444030, 26059512879605, 93952946906117
OFFSET
1,2
COMMENTS
A rooted tree is series-reduced if every non-leaf node has at least two branches.
In an identity tree, all branches directly under any given node are different.
LINKS
EXAMPLE
The a(1) = 1 through a(5) = 19 rooted trees:
(1) (2) (3) (4) (5)
(11) (111) (22) (11111)
((1)(2)) (1111) ((1)(4))
((1)(11)) ((1)(3)) ((2)(3))
((2)(11)) ((1)(22))
((1)(111)) ((3)(11))
((1)((1)(2))) ((2)(111))
((1)((1)(11))) ((1)(1111))
((11)(111))
((1)(2)(11))
((1)((1)(3)))
((2)((1)(2)))
((11)((1)(2)))
((1)((2)(11)))
((2)((1)(11)))
((1)((1)(111)))
((11)((1)(11)))
((1)((1)((1)(2))))
((1)((1)((1)(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]]]];
gob[m_]:=If[SameQ@@m, Prepend[#, m], #]&[Join@@Table[Select[Union[Sort/@Tuples[gob/@p]], UnsameQ@@#&], {p, Select[mps[m], Length[#]>1&]}]];
Table[Length[Join@@Table[gob[m], {m, IntegerPartitions[n]}]], {n, 10}]
PROG
(PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
seq(n)={my(v=vector(n)); for(n=1, n, v[n]=numdiv(n) + WeighT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 07 2018
EXTENSIONS
Terms a(13) and beyond from Andrew Howroyd, Oct 25 2018
STATUS
approved