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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A rooted tree is series-reduced if every non-leaf node has at least two branches.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..200

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

CROSSREFS

Cf. A000081, A000311, A000669, A001678, A005804, A141268, A292504, A300660, A317099, A319312, A320173, A320175.

Sequence in context: A148571 A345400 A148572 * A248603 A332344 A345244

Adjacent sequences:  A320171 A320172 A320173 * A320175 A320176 A320177

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 07 2018

EXTENSIONS

Terms a(11) and beyond from Andrew Howroyd, Oct 25 2018

STATUS

approved

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Last modified October 23 19:26 EDT 2021. Contains 348215 sequences. (Running on oeis4.)