A320177 Number of series-reduced rooted identity trees whose leaves are strict integer partitions whose multiset union is an integer partition of n.

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

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