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 A005805 Number of phylogenetic trees with n labels. (Formerly M1519) 3
 1, 2, 5, 18, 107, 1008, 13113, 214238, 4182487, 94747196, 2440784645, 70431957258, 2249856084803, 78802876705608, 3002702793753489, 123649410977736950, 5471808106109912815, 258948617502187143188, 13049542794706527317597, 697673361673877090147490 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Stirling transform of [ 1, 1, 1, 4, 26, 236, ... ] = A000311(n-1). Series-reduced trees where each leaf is a nonempty subset of the set of n labels. [Christian G. Bower, Dec 15 1999] 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..381 (first 100 terms from Vincenzo Librandi) N. J. A. Sloane, Transforms MAPLE stirtr:= proc(p) proc(n) add(p(k) *Stirling2(n, k), k=0..n) end end: b:= proc(n) option remember; if n<=1 then n elif n=2 then 1 else (n+1) *b(n-1) +2*add(binomial(n-1, k) *b(k) *b(n-k), k=2..n-2) fi end: a:= stirtr(n->`if`(n<2, 1, b(n-1))): seq(a(n), n=1..20);  # Alois P. Heinz, Sep 15 2008 MATHEMATICA max = 18; a311 = CoefficientList[ InverseSeries[ Series[ 1 + 2x - E^x, {x, 0, max}], x], x]*Range[0, max]!; b[1] = 1; b[k_] := a311[[k]]; a[n_] := Sum[ b[k]*StirlingS2[n, k], {k, 1, n}]; Table[ a[n], {n, 1, max}] (* Jean-François Alcover, Feb 22 2012 *) CROSSREFS Cf. A000311, A005804. Sequence in context: A304918 A007769 A174122 * A058338 A006896 A125625 Adjacent sequences:  A005802 A005803 A005804 * A005806 A005807 A005808 KEYWORD nonn,easy AUTHOR EXTENSIONS More terms from Christian G. Bower, Dec 15 1999 STATUS approved

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Last modified October 21 06:25 EDT 2019. Contains 328292 sequences. (Running on oeis4.)