A319312 - OEIS

%I #7 Sep 18 2018 17:08:08

%S 1,3,7,22,67,242,885,3456,13761,56342,234269,989335,4225341,18231145,

%T 79321931,347676128,1533613723,6803017863,30328303589,135808891308,

%U 610582497919,2755053631909,12472134557093,56630659451541,257841726747551,1176927093597201

%N Number of series-reduced rooted trees whose leaves are integer partitions whose multiset union is an integer partition of n.

%C Also the number of orderless tree-factorizations of Heinz numbers of integer partitions of n.

%C Also the number of phylogenetic trees on a multiset of labels summing to n.

%H Andrew Howroyd, <a href="/A319312/b319312.txt">Table of n, a(n) for n = 1..200</a>

%e The a(3) = 7 trees:

%e   (3)    (21)        (111)

%e        ((1)(2))    ((1)(11))

%e                   ((1)(1)(1))

%e                  ((1)((1)(1)))

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t phyfacs[n_]:=Prepend[Join@@Table[Union[Sort/@Tuples[phyfacs/@f]],{f,Select[facs[n],Length[#]>1&]}],n];

%t Table[Sum[Length[phyfacs[Times@@Prime/@m]],{m,IntegerPartitions[n]}],{n,6}]

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

%o seq(n)={my(v=[]); for(n=1, n, v=concat(v, numbpart(n) + EulerT(concat(v,[0]))[n])); v} \\ _Andrew Howroyd_, Sep 18 2018

%Y Cf. A000081, A000311, A000669, A001678, A005804, A141268, A292504, A300660, A316653, A316654, A316656.

%K nonn

%O 1,2

%A _Gus Wiseman_, Sep 17 2018

%E Terms a(14) and beyond from _Andrew Howroyd_, Sep 18 2018