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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.

%I A320294 #9 Oct 25 2018 22:21:51
%S A320294 0,0,0,1,1,3,3,7,8,15,19,37,48,87,126,227,342,611,964,1719,2806,4975,
%T A320294 8327,14782,25157,44609,76972,136622,237987,422881,742149,1320825,
%U A320294 2331491,4156392,7370868,13164429,23433637,41928557,74871434,134203411,240284935,431437069
%N 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.
%C A320294 Also phylogenetic trees with no singleton leaves on integer partitions of n with no 1's.
%H A320294 Andrew Howroyd, <a href="/A320294/b320294.txt">Table of n, a(n) for n = 1..500</a>
%e A320294 The a(4) = 1 through a(10) = 15 trees:
%e A320294   (22)  (32)  (33)   (43)   (44)        (54)        (55)
%e A320294               (42)   (52)   (53)        (63)        (64)
%e A320294               (222)  (322)  (62)        (72)        (73)
%e A320294                             (332)       (333)       (82)
%e A320294                             (422)       (432)       (433)
%e A320294                             (2222)      (522)       (442)
%e A320294                             ((22)(22))  (3222)      (532)
%e A320294                                         ((22)(23))  (622)
%e A320294                                                     (3322)
%e A320294                                                     (4222)
%e A320294                                                     (22222)
%e A320294                                                     ((22)(24))
%e A320294                                                     ((22)(33))
%e A320294                                                     ((23)(23))
%e A320294                                                     ((22)(222))
%t A320294 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A320294 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A320294 pgtm[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[pgtm/@p]],{p,Select[mps[m],Length[#]>1&]}],m];
%t A320294 Table[Sum[Length[Select[pgtm[m],FreeQ[#,{_}]&]],{m,Select[IntegerPartitions[n],FreeQ[#,1]&]}],{n,10}]
%o A320294 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A320294 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 A320294 Cf. A000045, A000311, A000669, A002865, A141268, A292504, A304966, A304967, A319312, A320289, A320293, A320295, A320296.
%K A320294 nonn
%O A320294 1,6
%A A320294 _Gus Wiseman_, Oct 09 2018
%E A320294 Terms a(16) and beyond from _Andrew Howroyd_, Oct 25 2018