

A007827


Number of homeomorphically irreducible (or seriesreduced) trees with n pendant nodes, or continua with n noncut 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
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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), 177184.
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.
Index entries for sequences related to trees
Index entries for sequences related to rooted trees


FORMULA

G.f.: 1+(1+xB(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+xB)*B, x, 30); # where B = g.f. for A000669; A007827 := n>coeff(A, x, n);


MATHEMATICA

(* a9 = A000669 *) max = 29; a9[1] = 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] (* JeanFrançois Alcover, Aug 14 2012 *)


CROSSREFS

Cf. A000014 (seriesreduced trees), A000055 (trees), A000311, A000669 (seriesreduced planted trees by leaves), A059123 (homeomorphically irreducible rooted trees by nodes), A271205 (seriesreduced 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



