A320296 Number of series-reduced rooted trees whose leaves form an integer partition of n with no 1's.

0, 1, 1, 2, 2, 5, 6, 15, 22, 51, 86, 195, 354, 781, 1512, 3286, 6602, 14269, 29424, 63494, 133298, 287909, 612188, 1325375, 2844448, 6176145, 13348858, 29074164, 63187176, 138044144, 301350424, 660265471, 1446678326, 3178246273, 6985464590, 15384556290

Offset

1,4

Comments

Also phylogenetic trees with n unlabeled objects and no singleton leaves.

Links

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

Example

The a(2) = 1 through a(9) = 22 trees:
2   3   4     5     6        7        8           9
        (22)  (23)  (24)     (25)     (26)        (27)
                    (33)     (34)     (35)        (36)
                    (222)    (223)    (44)        (45)
                    (2(22))  ((22)3)  (224)       (225)
                             (2(23))  (233)       (234)
                                      (2222)      (333)
                                      ((22)4)     (2223)
                                      (2(24))     ((22)5)
                                      ((23)3)     (2(25))
                                      (2(33))     ((23)4)
                                      (2(222))    (2(34))
                                      (22(22))    ((24)3)
                                      ((22)(22))  ((33)3)
                                      (2(2(22)))  (2(22)3)
                                                  (2(223))
                                                  (22(23))
                                                  (3(222))
                                                  ((2(22))3)
                                                  ((22)(23))
                                                  (2((22)3))
                                                  (2(2(23)))

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
t[n_]:=t[n]=If[PrimeQ[n], {n}, Join@@Table[Union[Sort/@Tuples[t/@fac]], {fac, Select[facs[n], Length[#]>1&]}]];
Table[Sum[Length[t[Times@@Prime/@ptn]], {ptn, Select[IntegerPartitions[n], FreeQ[#, 1]&]}], {n, 15}]

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=2, n, v[n]=1 + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Crossrefs

Cf. A000311, A000669, A002865, A005804, A141268, A304967, A319312, A320293.

Sequence in context: A050216 A345470 A368727 * A329908 A295198 A080880

Adjacent sequences: A320293 A320294 A320295 * A320297 A320298 A320299

Keyword

nonn

Author

Gus Wiseman, Oct 09 2018

Extensions

Terms a(26) and beyond from Andrew Howroyd, Oct 25 2018

Status

approved