A320175 Number of series-reduced rooted trees whose leaves are strict integer partitions whose multiset union is an integer partition of n.

1, 2, 5, 13, 37, 120, 395, 1381, 4931, 18074, 67287, 254387, 972559, 3756315, 14629237, 57395490, 226613217, 899773355, 3590349661, 14390323014, 57907783039, 233867667197, 947601928915, 3851054528838, 15693587686823, 64114744713845, 262543966114921, 1077406218930902

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) = 13 trees:
  (1)  (2)       (3)            (4)
       ((1)(1))  (21)           (31)
                 ((1)(2))       ((1)(3))
                 ((1)(1)(1))    ((2)(2))
                 ((1)((1)(1)))  ((1)(21))
                                ((1)(1)(2))
                                ((1)((1)(2)))
                                ((2)((1)(1)))
                                ((1)(1)(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]]]];
sot[m_]:=If[UnsameQ@@m, Prepend[#, m], #]&[Join@@Table[Union[Sort/@Tuples[sot/@p]], {p, Select[mps[m], Length[#]>1&]}]];
Table[Length[Join@@Table[sot[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(p=prod(k=1, n, 1 + x^k + O(x*x^n)), v=vector(n)); for(n=1, n, v[n]=polcoef(p, 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, A320174.

Sequence in context: A205544 A149855 A149856 * A064384 A148302 A149857

Adjacent sequences: A320172 A320173 A320174 * A320176 A320177 A320178

Keyword

nonn

Author

Gus Wiseman, Oct 07 2018

Extensions

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

Status

approved