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A005804 Number of phylogenetic rooted trees with n labels.
(Formerly M1890)

%I M1890

%S 1,2,8,58,612,8374,140408,2785906,63830764,1658336270,48169385024,

%T 1546832023114,54413083601268,2080827594898342,85948745163598088,

%U 3813417859420469410,180876816831806597500,9133309115320844870078,489156459621633161274704,27696066472039561313329018

%N Number of phylogenetic rooted trees with n labels.

%C These are series-reduced rooted trees where each leaf is a nonempty subset of the set of n labels.

%C See A141268 for phylogenetic rooted trees with n unlabeled objects. - _Thomas Wieder_, Jun 20 2008

%D Foulds, L. R.; Robinson, R. W. Enumeration of phylogenetic trees without points of degree two. Ars Combin. 17 (1984), A, 169-183. Math. Rev. 85f:05045

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A005804/b005804.txt">Table of n, a(n) for n = 1..380</a> (first 100 terms from T. D. Noe)

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F Stirling transform of [ 1, 1, 4, 26, 236, ... ] = A000311 [ Foulds and Robinson ].

%F E.g.f.: -LambertW(-1/2*exp(1/2*exp(z)-1))+1/2*exp(z)-1 series(-LambertW(-1/2*exp(1/2*exp(z)-1))+1/2*exp(z)-1,z=0,10). - _Thomas Wieder_, Jun 20 2008

%F a(n) ~ sqrt(log(2))*(log(2)+log(log(2)))^(1/2-n)*n^(n-1)/exp(n). - _Vaclav Kotesovec_, Aug 07 2013

%F E.g.f. f(x) satisfies 2 f(x) - exp(f(x)) = exp(x) - 1. - _Gus Wiseman_, Jul 31 2018

%e a(3)=8 because we have:

%e Set(Set(Z[3]),Set(Z[1]),Set(Z[2])),

%e Set(Z[3],Z[2],Z[1]),

%e Set(Set(Z[3],Z[1]),Set(Z[2])),

%e Set(Set(Set(Z[3]),Set(Z[2])),Set(Z[1])),

%e Set(Set(Set(Z[3]),Set(Z[1])),Set(Z[2])),

%e Set(Set(Z[3]),Set(Set(Z[1]),Set(Z[2]))),

%e Set(Set(Z[3]),Set(Z[2],Z[1])),

%e Set(Set(Z[3],Z[2]),Set(Z[1]))

%e From _Gus Wiseman_, Jul 31 2018: (Start)

%e The 8 series-reduced rooted trees whose leaves are a set partition of {1,2,3}:

%e {1,2,3}

%e ({1}{2,3})

%e ({1}({2}{3}))

%e ({2}{1,3})

%e ({2}({1}{3}))

%e ({3}{1,2})

%e ({3}({1}{2}))

%e ({1}{2}{3})

%e (End)

%p with(combstruct): A005804 := [H, {H=Union(Set(Z,card>=1),Set(H,card>=2))}, labelled]; seq(count(A005804,size=j),j=1..20); # _Thomas Wieder_, Jun 20 2008

%t numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];

%t a[n_]:=a[n]=If[n==1,1,1+Sum[numSetPtnsOfType[ptn]*Times@@a/@ptn,{ptn,Rest[IntegerPartitions[n]]}]];

%t Array[a,20] (* _Gus Wiseman_, Jul 31 2018 *)

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

%o b(n,k)={my(v=vector(n)); for(n=1, n, v[n]=binomial(n+k-1, n) + EulerT(v[1..n])[n]); v}

%o 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 Cf. A000081, A000311, A000669, A001678, A005805, A141268, A292504, A300660, A316656.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_

%E More terms, comment from _Christian G. Bower_, Dec 15 1999

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Last modified February 18 10:13 EST 2019. Contains 320250 sequences. (Running on oeis4.)