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A320174
Number of series-reduced rooted trees whose leaves are constant integer partitions whose multiset union is an integer partition of n.
6
1, 3, 6, 19, 55, 200, 713, 2740, 10651, 42637, 173012, 713280, 2972389, 12514188, 53119400, 227140464, 977382586, 4229274235, 18391269922, 80330516578, 352269725526, 1550357247476, 6845517553493, 30316222112019, 134626183784975, 599341552234773, 2674393679352974
OFFSET
1,2
COMMENTS
A rooted tree is series-reduced if every non-leaf node has at least two branches.
LINKS
EXAMPLE
The a(1) = 1 through a(4) = 19 trees:
(1) (2) (3) (4)
(11) (111) (22)
((1)(1)) ((1)(2)) (1111)
((1)(11)) ((1)(3))
((1)(1)(1)) ((2)(2))
((1)((1)(1))) ((2)(11))
((1)(111))
((11)(11))
((1)(1)(2))
((1)(1)(11))
((1)((1)(2)))
((2)((1)(1)))
((1)((1)(11)))
((1)(1)(1)(1))
((11)((1)(1)))
((1)((1)(1)(1)))
((1)(1)((1)(1)))
(((1)(1))((1)(1)))
((1)((1)((1)(1))))
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]]]];
dot[m_]:=If[SameQ@@m, Prepend[#, m], #]&[Join@@Table[Union[Sort/@Tuples[dot/@p]], {p, Select[mps[m], Length[#]>1&]}]];
Table[Length[Join@@Table[dot[m], {m, IntegerPartitions[n]}]], {n, 10}]
PROG
(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=vector(n)); for(n=1, n, v[n]=numdiv(n) + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 07 2018
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Oct 25 2018
STATUS
approved