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 A007827 Number of homeomorphically irreducible (or series-reduced) trees with n pendant nodes, or continua with n non-cut points, or leaves. 10
 1, 1, 1, 1, 2, 3, 7, 13, 32, 73, 190, 488, 1350, 3741, 10765, 31311, 92949, 278840, 847511, 2599071, 8044399, 25082609, 78758786, 248803504, 790411028, 2523668997, 8095146289, 26076714609, 84329102797, 273694746208 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Also, number of unrooted multifurcating tree shapes with n leaves [see Felsenstein]. REFERENCES M. Cropper, J. Combin. Math. Combin. Comp., Vol. 24 (1997), 177-184. Joseph Felsenstein, Inferring Phylogenies. Sinauer Associates, Inc., 2004, p. 33 (Beware errors!). F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 62. S. B. Nadler Jr., Continuum Theory, Academic Press. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..100 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. FORMULA G.f.: 1+(1+x-B(x))*B(x) where B(x) = x+x^2+2*x^3+5*x^4+12*x^5+33*x^6+90*x^7+... is g.f. for A000669. MAPLE A := series(1+(1+x-B)*B, x, 30); # where B = g.f. for A000669; A007827 := n->coeff(A, x, n); MATHEMATICA (* a9 = A000669 *) max = 29; a9 = 1; a9[n_] := (s = Series[1/(1 - x), {x, 0, n}]; Do[s = Series[s/(1 - x^k)^Coefficient[s, x^k], {x, 0, n}], {k, 2, n}]; Coefficient[s, x^n]/2); b[x_] := Sum[a9[n] x^n, {n, 1, max}]; gf[x_] := 1 + (1 + x - b[x])*b[x]; CoefficientList[ Series[gf[x], {x, 0, max}], x] (* Jean-François Alcover, Aug 14 2012 *) CROSSREFS Cf. A000014 (series-reduced trees), A000055 (trees), A000311, A000669 (series-reduced planted trees by leaves), A059123 (homeomorphically irreducible rooted trees by nodes), A271205 (series-reduced trees by leaves and nodes). Number of row entries of A064060. Sequence in context: A003120 A032131 A324844 * A250308 A259145 A237255 Adjacent sequences:  A007824 A007825 A007826 * A007828 A007829 A007830 KEYWORD nonn,nice,easy AUTHOR Matthew Cropper (mmcrop01(AT)athena.louisville.edu). EXTENSIONS Corrected and extended by Christian G. Bower, Nov 15 1999 STATUS approved

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Last modified February 28 04:39 EST 2020. Contains 332321 sequences. (Running on oeis4.)