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A320171 Number of series-reduced rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

%I A320171 #8 Oct 26 2018 00:52:27
%S A320171 1,2,5,11,29,82,247,782,2579,8702,29975,104818,371111,1327307,4788687,
%T A320171 17404838,63669763,234237605,866090021,3216738344,11995470691,
%U A320171 44894977263,168582174353,634939697164,2398004674911,9079614633247,34458722286825,131059771522401
%N A320171 Number of series-reduced rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.
%C A320171 A rooted tree is series-reduced if every non-leaf node has at least two branches.
%C A320171 In an identity tree, all branches directly under any given node are different.
%H A320171 Andrew Howroyd, <a href="/A320171/b320171.txt">Table of n, a(n) for n = 1..200</a>
%e A320171 The a(1) = 1 through a(4) = 11 rooted identity trees:
%e A320171   (1)  (2)   (3)        (4)
%e A320171        (11)  (21)       (22)
%e A320171              (111)      (31)
%e A320171              ((1)(2))   (211)
%e A320171              ((1)(11))  (1111)
%e A320171                         ((1)(3))
%e A320171                         ((1)(21))
%e A320171                         ((2)(11))
%e A320171                         ((1)(111))
%e A320171                         ((1)((1)(2)))
%e A320171                         ((1)((1)(11)))
%t A320171 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A320171 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A320171 gig[m_]:=Prepend[Join@@Table[Union[Sort/@Select[Sort/@Tuples[gig/@mtn],UnsameQ@@#&]],{mtn,Select[mps[m],Length[#]>1&]}],m];
%t A320171 Table[Sum[Length[gig[y]],{y,IntegerPartitions[n]}],{n,8}]
%o A320171 (PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
%o A320171 seq(n)={my(v=vector(n)); for(n=1, n, v[n]=numbpart(n) + WeighT(v[1..n])[n]); v} \\ _Andrew Howroyd_, Oct 25 2018
%Y A320171 Cf. A000081, A000311, A000669, A001678, A005804, A141268, A292504, A300660, A319312, A320172, A320177, A320178.
%K A320171 nonn
%O A320171 1,2
%A A320171 _Gus Wiseman_, Oct 07 2018
%E A320171 Terms a(12) and beyond from _Andrew Howroyd_, Oct 25 2018