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%I #10 Oct 25 2018 22:20:44
%S 1,1,3,5,11,26,65,169,463,1294,3691,10700,31417,93175,278805,840424,
%T 2549895,7780472,23860359,73500838,227330605,705669634,2197750615,
%U 6865335389,21505105039,67533738479,212575923471,670572120240,2119568530289,6712115439347
%N Number of series-reduced rooted identity trees whose leaves are strict 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="/A320177/b320177.txt">Table of n, a(n) for n = 1..200</a>
%e The a(1) = 1 through a(5) = 11 rooted trees:
%e (1) (2) (3) (4) (5)
%e (21) (31) (32)
%e ((1)(2)) ((1)(3)) (41)
%e ((1)(12)) ((1)(4))
%e ((1)((1)(2))) ((2)(3))
%e ((1)(13))
%e ((2)(12))
%e ((1)((1)(3)))
%e ((2)((1)(2)))
%e ((1)((1)(12)))
%e ((1)((1)((1)(2))))
%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 gog[m_]:=If[UnsameQ@@m,Prepend[#,m],#]&[Join@@Table[Select[Union[Sort/@Tuples[gog/@p]],UnsameQ@@#&],{p,Select[mps[m],Length[#]>1&]}]];
%t Table[Length[Join@@Table[gog[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(p=prod(k=1, n, 1 + x^k + O(x*x^n)), v=vector(n)); for(n=1, n, v[n]=polcoef(p, 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, A320178.
%K nonn
%O 1,3
%A _Gus Wiseman_, Oct 07 2018
%E Terms a(13) and beyond from _Andrew Howroyd_, Oct 25 2018
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