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A000014
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Number of series-reduced trees with n nodes.
(Formerly M0320 N0118)
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23
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0, 1, 1, 0, 1, 1, 2, 2, 4, 5, 10, 14, 26, 42, 78, 132, 249, 445, 842, 1561, 2988, 5671, 10981, 21209, 41472, 81181, 160176, 316749, 629933, 1256070, 2515169, 5049816, 10172638, 20543579, 41602425, 84440886, 171794492, 350238175, 715497037, 1464407113
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OFFSET
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0,7
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COMMENTS
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Other terms for "series-reduced tree": (i) homeomorphically irreducible tree, (ii) homeomorphically reduced tree, (iii) reduced tree, (iv) topological tree.
In a series-reduced tree, vertices cannot have degree 2; they can be leaves or have >= 2 branches.
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REFERENCES
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F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 284.
D. G. Cantor, personal communication.
F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 232.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 62, Fig. 3.3.3.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 526.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Peter Steinbach, Field Guide to Simple Graphs, Volume 3, Part 12 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
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FORMULA
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G.f.: A(x) = ((x-1)/x)*f(x) + ((1+x)/x^2)*g(x) - (1/x^2)*g(x)^2 where f(x) is g.f. for A059123 and g(x) is g.f. for A001678. [Harary and E. M. Palmer, p. 62, Eq. (3.3.10) with extra -(1/x^2)*Hbar(x)^2 term which should be there according to eq.(3.3.14), p. 63, with eq.(3.3.9)]. [corrected by Wolfdieter Lang, Jan 09 2001]
a(n) ~ c * d^n / n^(5/2), where d = A246403 = 2.189461985660850..., c = 0.684447272004914061023163279794145361469033868145768075109924585532604582794... - Vaclav Kotesovec, Aug 25 2014
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EXAMPLE
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G.f. = x + x^2 + x^4 + x^5 + 2*x^6 + 2*x^7 + 4*x^8 + 5*x^9 + 10*x^10 + ...
The star graph with n nodes (except for n=3) is a series-reduced tree. For n=6 the other series-reduced tree is shaped like the letter H. - Michael Somos, Dec 19 2014
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MAPLE
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with(powseries): with(combstruct): n := 30: Order := n+3: sys := {B = Prod(C, Z), S = Set(B, 1 <= card), C = Union(Z, S)}:
G001678 := (convert(gfseries(sys, unlabeled, x) [S(x)], polynom)) * x^2: G0temp := G001678 + x^2:
G059123 := G0temp / x + G0temp - (G0temp^2+eval(G0temp, x=x^2))/(2*x):
G000014 := ((x-1)/x) * G059123 + ((1+x)/x^2) * G0temp - (1/x^2) * G0temp^2:
A000014 := 0, seq(coeff(G000014, x^i), i=1..n); # Ulrich Schimke (ulrschimke(AT)aol.com)
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MATHEMATICA
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a[n_] := If[n<1, 0, A = x/(1-x^2) + x*O[x]^n; For[k=3, k <= n-1, k++, A = A/(1 - x^k + x*O[x]^n)^SeriesCoefficient[A, k]]; s = ((Normal[A] /. x -> x^2) + O[x]^(2n))*(1-x) + A*(2-A)*(1+x); SeriesCoefficient[s, n]/2]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 02 2016, adapted from PARI *)
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PROG
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(PARI) {a(n) = my(A); if( n<1, 0, A = x / (1 - x^2) + x * O(x^n); for(k=3, n-1, A /= (1 - x^k + x * O(x^n))^polcoeff(A, k)); polcoeff( (subst(A, x, x^2) * (1 - x) + A * (2 - A) * (1 + x)) / 2, n))}; /* Michael Somos, Dec 19 2014 */
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CROSSREFS
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Cf. A000055 (trees), A001678 (series-reduced planted trees), A007827 (series-reduced trees by leaves), A271205 (series-reduced trees by leaves and nodes).
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KEYWORD
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nonn,easy,core,nice
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AUTHOR
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STATUS
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approved
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