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

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Last modified October 15 00:14 EDT 2019. Contains 328025 sequences. (Running on oeis4.)