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

%I A320296 #7 Oct 26 2018 00:52:51
%S A320296 0,1,1,2,2,5,6,15,22,51,86,195,354,781,1512,3286,6602,14269,29424,
%T A320296 63494,133298,287909,612188,1325375,2844448,6176145,13348858,29074164,
%U A320296 63187176,138044144,301350424,660265471,1446678326,3178246273,6985464590,15384556290
%N A320296 Number of series-reduced rooted trees whose leaves form an integer partition of n with no 1's.
%C A320296 Also phylogenetic trees with n unlabeled objects and no singleton leaves.
%H A320296 Andrew Howroyd, <a href="/A320296/b320296.txt">Table of n, a(n) for n = 1..500</a>
%e A320296 The a(2) = 1 through a(9) = 22 trees:
%e A320296 2   3   4     5     6        7        8           9
%e A320296         (22)  (23)  (24)     (25)     (26)        (27)
%e A320296                     (33)     (34)     (35)        (36)
%e A320296                     (222)    (223)    (44)        (45)
%e A320296                     (2(22))  ((22)3)  (224)       (225)
%e A320296                              (2(23))  (233)       (234)
%e A320296                                       (2222)      (333)
%e A320296                                       ((22)4)     (2223)
%e A320296                                       (2(24))     ((22)5)
%e A320296                                       ((23)3)     (2(25))
%e A320296                                       (2(33))     ((23)4)
%e A320296                                       (2(222))    (2(34))
%e A320296                                       (22(22))    ((24)3)
%e A320296                                       ((22)(22))  ((33)3)
%e A320296                                       (2(2(22)))  (2(22)3)
%e A320296                                                   (2(223))
%e A320296                                                   (22(23))
%e A320296                                                   (3(222))
%e A320296                                                   ((2(22))3)
%e A320296                                                   ((22)(23))
%e A320296                                                   (2((22)3))
%e A320296                                                   (2(2(23)))
%t A320296 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A320296 t[n_]:=t[n]=If[PrimeQ[n],{n},Join@@Table[Union[Sort/@Tuples[t/@fac]],{fac,Select[facs[n],Length[#]>1&]}]];
%t A320296 Table[Sum[Length[t[Times@@Prime/@ptn]],{ptn,Select[IntegerPartitions[n],FreeQ[#,1]&]}],{n,15}]
%o A320296 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A320296 seq(n)={my(v=vector(n)); for(n=2, n, v[n]=1 + EulerT(v[1..n])[n]); v} \\ _Andrew Howroyd_, Oct 25 2018
%Y A320296 Cf. A000311, A000669, A002865, A005804, A141268, A304967, A319312, A320293.
%K A320296 nonn
%O A320296 1,4
%A A320296 _Gus Wiseman_, Oct 09 2018
%E A320296 Terms a(26) and beyond from _Andrew Howroyd_, Oct 25 2018