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A320177 Number of series-reduced rooted identity trees whose leaves are strict integer partitions whose multiset union is an integer partition of n. 5
1, 1, 3, 5, 11, 26, 65, 169, 463, 1294, 3691, 10700, 31417, 93175, 278805, 840424, 2549895, 7780472, 23860359, 73500838, 227330605, 705669634, 2197750615, 6865335389, 21505105039, 67533738479, 212575923471, 670572120240, 2119568530289, 6712115439347 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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

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

EXAMPLE

The a(1) = 1 through a(5) = 11 rooted trees:

  (1)  (2)  (3)       (4)            (5)

            (21)      (31)           (32)

            ((1)(2))  ((1)(3))       (41)

                      ((1)(12))      ((1)(4))

                      ((1)((1)(2)))  ((2)(3))

                                     ((1)(13))

                                     ((2)(12))

                                     ((1)((1)(3)))

                                     ((2)((1)(2)))

                                     ((1)((1)(12)))

                                     ((1)((1)((1)(2))))

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]]]];

gog[m_]:=If[UnsameQ@@m, Prepend[#, m], #]&[Join@@Table[Select[Union[Sort/@Tuples[gog/@p]], UnsameQ@@#&], {p, Select[mps[m], Length[#]>1&]}]];

Table[Length[Join@@Table[gog[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(p=prod(k=1, n, 1 + x^k + O(x*x^n)), v=vector(n)); for(n=1, n, v[n]=polcoef(p, n) + WeighT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

CROSSREFS

Cf. A000669, A004111, A005804, A141268, A292504, A300660, A319312.

Cf. A320171, A320174, A320175, A320176, A320178.

Sequence in context: A240148 A109249 A196423 * A289468 A289534 A032364

Adjacent sequences:  A320174 A320175 A320176 * A320178 A320179 A320180

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 07 2018

EXTENSIONS

Terms a(13) and beyond from Andrew Howroyd, Oct 25 2018

STATUS

approved

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Last modified September 20 14:44 EDT 2021. Contains 347586 sequences. (Running on oeis4.)