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

Author

Gus Wiseman, Oct 09 2018

Keywords

• nonn

Comments

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

Examples

			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)))
		

Links

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

Crossrefs

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

Programs

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}]

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

Extensions

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