A320294 - OEIS

%I #9 Oct 25 2018 22:21:51

%S 0,0,0,1,1,3,3,7,8,15,19,37,48,87,126,227,342,611,964,1719,2806,4975,

%T 8327,14782,25157,44609,76972,136622,237987,422881,742149,1320825,

%U 2331491,4156392,7370868,13164429,23433637,41928557,74871434,134203411,240284935,431437069

%N 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.

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

%H Andrew Howroyd, <a href="/A320294/b320294.txt">Table of n, a(n) for n = 1..500</a>

%e The a(4) = 1 through a(10) = 15 trees:

%e   (22)  (32)  (33)   (43)   (44)        (54)        (55)

%e               (42)   (52)   (53)        (63)        (64)

%e               (222)  (322)  (62)        (72)        (73)

%e                             (332)       (333)       (82)

%e                             (422)       (432)       (433)

%e                             (2222)      (522)       (442)

%e                             ((22)(22))  (3222)      (532)

%e                                         ((22)(23))  (622)

%e                                                     (3322)

%e                                                     (4222)

%e                                                     (22222)

%e                                                     ((22)(24))

%e                                                     ((22)(33))

%e                                                     ((23)(23))

%e                                                     ((22)(222))

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

%t pgtm[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[pgtm/@p]],{p,Select[mps[m],Length[#]>1&]}],m];

%t Table[Sum[Length[Select[pgtm[m],FreeQ[#,{_}]&]],{m,Select[IntegerPartitions[n],FreeQ[#,1]&]}],{n,10}]

%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}

%o 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

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

%K nonn

%O 1,6

%A _Gus Wiseman_, Oct 09 2018

%E Terms a(16) and beyond from _Andrew Howroyd_, Oct 25 2018