A320294 Number of series-reduced rooted trees whose leaves are non-singleton integer partitions whose multiset union is an integer partition of n with no 1's.

0, 0, 0, 1, 1, 3, 3, 7, 8, 15, 19, 37, 48, 87, 126, 227, 342, 611, 964, 1719, 2806, 4975, 8327, 14782, 25157, 44609, 76972, 136622, 237987, 422881, 742149, 1320825, 2331491, 4156392, 7370868, 13164429, 23433637, 41928557, 74871434, 134203411, 240284935, 431437069

Offset

1,6

Comments

Also phylogenetic trees with no singleton leaves on integer partitions of n with no 1's.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..500

Example

The a(4) = 1 through a(10) = 15 trees:
  (22)  (32)  (33)   (43)   (44)        (54)        (55)
              (42)   (52)   (53)        (63)        (64)
              (222)  (322)  (62)        (72)        (73)
                            (332)       (333)       (82)
                            (422)       (432)       (433)
                            (2222)      (522)       (442)
                            ((22)(22))  (3222)      (532)
                                        ((22)(23))  (622)
                                                    (3322)
                                                    (4222)
                                                    (22222)
                                                    ((22)(24))
                                                    ((22)(33))
                                                    ((23)(23))
                                                    ((22)(222))

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]]]];
pgtm[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[pgtm/@p]], {p, Select[mps[m], Length[#]>1&]}], m];
Table[Sum[Length[Select[pgtm[m], FreeQ[#, {_}]&]], {m, Select[IntegerPartitions[n], FreeQ[#, 1]&]}], {n, 10}]

Prog

(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(p=1/prod(k=2, n, 1 - x^k + O(x*x^n)), v=vector(n)); for(n=2, n, v[n]=polcoef(p, n) - 1 + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Crossrefs

Cf. A000045, A000311, A000669, A002865, A141268, A292504, A304966, A304967, A319312, A320289, A320293, A320295, A320296.

Sequence in context: A281616 A110618 A320291 * A304179 A182559 A108046

Adjacent sequences: A320291 A320292 A320293 * A320295 A320296 A320297

Keyword

nonn

Author

Gus Wiseman, Oct 09 2018

Extensions

Terms a(16) and beyond from Andrew Howroyd, Oct 25 2018

Status

approved