A320178 - OEIS

%I #9 Oct 25 2018 22:21:00

%S 1,2,4,8,19,53,151,459,1445,4634,15154,50253,168607,571212,1951588,

%T 6715575,23255444,80978697,283373024,995995996,3514614634,12446666967,

%U 44222390525,157587392768,563096832839,2017121728223,7242436444030,26059512879605,93952946906117

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

%C A rooted tree is series-reduced if every non-leaf node has at least two branches.

%C In an identity tree, all branches directly under any given node are different.

%H Andrew Howroyd, <a href="/A320178/b320178.txt">Table of n, a(n) for n = 1..200</a>

%e The a(1) = 1 through a(5) = 19 rooted trees:

%e   (1)  (2)   (3)        (4)             (5)

%e        (11)  (111)      (22)            (11111)

%e              ((1)(2))   (1111)          ((1)(4))

%e              ((1)(11))  ((1)(3))        ((2)(3))

%e                         ((2)(11))       ((1)(22))

%e                         ((1)(111))      ((3)(11))

%e                         ((1)((1)(2)))   ((2)(111))

%e                         ((1)((1)(11)))  ((1)(1111))

%e                                         ((11)(111))

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

%e                                         ((1)((1)(3)))

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

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

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

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

%e                                         ((1)((1)(111)))

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

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

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

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

%t Table[Length[Join@@Table[gob[m],{m,IntegerPartitions[n]}]],{n,10}]

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

%o seq(n)={my(v=vector(n)); for(n=1, n, v[n]=numdiv(n) + WeighT(v[1..n])[n]); v} \\ _Andrew Howroyd_, Oct 25 2018

%Y Cf. A000669, A004111, A005804, A141268, A292504, A300660, A319312.

%Y Cf. A320171, A320174, A320175, A320176, A320177.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 07 2018

%E Terms a(13) and beyond from _Andrew Howroyd_, Oct 25 2018