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A320154 Number of series-reduced balanced rooted trees whose leaves form a set partition of {1,...,n}.

%I A320154 #8 Oct 26 2018 17:11:28
%S A320154 1,2,5,18,92,588,4328,35920,338437,3654751,45105744,625582147,
%T A320154 9539374171,157031052142,2757275781918,51293875591794,
%U A320154 1007329489077804,20840741773898303,453654220906310222,10380640686263467204,249559854371799622350,6301679967177242849680
%N A320154 Number of series-reduced balanced rooted trees whose leaves form a set partition of {1,...,n}.
%C A320154 A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root.
%C A320154 Also the number of balanced phylogenetic rooted trees on n distinct labels.
%H A320154 Andrew Howroyd, <a href="/A320154/b320154.txt">Table of n, a(n) for n = 1..200</a>
%e A320154 The a(1) = 1 through a(4) = 18 rooted trees:
%e A320154   (1)  (12)      (123)        (1234)
%e A320154        ((1)(2))  ((1)(23))    ((1)(234))
%e A320154                  ((2)(13))    ((12)(34))
%e A320154                  ((3)(12))    ((13)(24))
%e A320154                  ((1)(2)(3))  ((14)(23))
%e A320154                               ((2)(134))
%e A320154                               ((3)(124))
%e A320154                               ((4)(123))
%e A320154                               ((1)(2)(34))
%e A320154                               ((1)(3)(24))
%e A320154                               ((1)(4)(23))
%e A320154                               ((2)(3)(14))
%e A320154                               ((2)(4)(13))
%e A320154                               ((3)(4)(12))
%e A320154                               ((1)(2)(3)(4))
%e A320154                               (((1)(2))((3)(4)))
%e A320154                               (((1)(3))((2)(4)))
%e A320154                               (((1)(4))((2)(3)))
%t A320154 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A320154 gug[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[gug/@mtn]],{mtn,Select[sps[m],Length[#]>1&]}],m];
%t A320154 Table[Length[Select[gug[Range[n]],SameQ@@Length/@Position[#,_Integer]&]],{n,9}]
%o A320154 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A320154 b(n,k)={my(u=vector(n), v=vector(n)); u[1]=k; u=EulerT(u); while(u, v+=u; u=EulerT(u)-u); v}
%o A320154 seq(n)={my(M=Mat(vectorv(n,k,b(n,k)))); vector(n, k, sum(i=1, k, binomial(k,i)*(-1)^(k-i)*M[i,k]))} \\ _Andrew Howroyd_, Oct 26 2018
%Y A320154 Cf. A000081, A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A244925, A292504, A300660, A319312.
%Y A320154 Cf. A320155, A320160, A316624, A320169, A320173, A320176, A320179.
%K A320154 nonn
%O A320154 1,2
%A A320154 _Gus Wiseman_, Oct 06 2018
%E A320154 Terms a(9) and beyond from _Andrew Howroyd_, Oct 26 2018