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A005804 Number of phylogenetic rooted trees with n labels.
(Formerly M1890)
36
1, 2, 8, 58, 612, 8374, 140408, 2785906, 63830764, 1658336270, 48169385024, 1546832023114, 54413083601268, 2080827594898342, 85948745163598088, 3813417859420469410, 180876816831806597500, 9133309115320844870078, 489156459621633161274704, 27696066472039561313329018 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

See A141268 for phylogenetic rooted trees with n unlabeled objects. - Thomas Wieder, Jun 20 2008

REFERENCES

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

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..380 (first 100 terms from T. D. Noe)

N. J. A. Sloane, Transforms

Index entries for sequences related to rooted trees

FORMULA

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

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

a(n) ~ sqrt(log(2))*(log(2)+log(log(2)))^(1/2-n)*n^(n-1)/exp(n). - Vaclav Kotesovec, Aug 07 2013

E.g.f. f(x) satisfies  2 f(x) - exp(f(x)) = exp(x) - 1. - Gus Wiseman, Jul 31 2018

EXAMPLE

a(3)=8 because we have:

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

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

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

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

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

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

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

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

From Gus Wiseman, Jul 31 2018: (Start)

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

  {1,2,3}

  ({1}{2,3})

  ({1}({2}{3}))

  ({2}{1,3})

  ({2}({1}{3}))

  ({3}{1,2})

  ({3}({1}{2}))

  ({1}{2}{3})

(End)

MAPLE

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

MATHEMATICA

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

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

Array[a, 20] (* Gus Wiseman, Jul 31 2018 *)

PROG

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

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

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

CROSSREFS

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

Sequence in context: A185898 A063074 A319590 * A162067 A179534 A256034

Adjacent sequences:  A005801 A005802 A005803 * A005805 A005806 A005807

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms, comment from Christian G. Bower, Dec 15 1999

STATUS

approved

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Last modified November 13 21:28 EST 2019. Contains 329106 sequences. (Running on oeis4.)