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 A059123 Number of homeomorphically irreducible rooted trees (also known as series-reduced rooted trees, or rooted trees without nodes of degree 2) with n >= 1 nodes. 6
 0, 1, 1, 0, 2, 2, 4, 6, 12, 20, 39, 71, 137, 261, 511, 995, 1974, 3915, 7841, 15749, 31835, 64540, 131453, 268498, 550324, 1130899, 2330381, 4813031, 9963288, 20665781, 42947715, 89410092, 186447559, 389397778, 814447067, 1705775653 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 REFERENCES F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 62, Eq. (3.3.9). LINKS N. J. A. Sloane, Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014) P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183. N. J. A. Sloane, Illustration of initial terms FORMULA G.f.: 1 + ((1+x)/x)*f(x) - (f(x)^2+f(x^2))/(2*x) where 1+f(x) is g.f. for A001678 (homeomorphically irreducible planted trees by nodes). a(n) = A001679(n) if n>0. - Michael Somos, Jun 13 2014 a(n) ~ c * d^n / n^(3/2), where d = A246403 = 2.18946198566085056388702757711... and c = 0.4213018528699249210965028... . - Vaclav Kotesovec, Jun 26 2014 EXAMPLE G.f. = x + x^2 + 2*x^4 + 2*x^5 + 4*x^6 + 6*x^7 + 12*x^8 + 20*x^9 + ... MAPLE 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): A059123 := 0, seq(coeff(G059123, x^i), i=1..n); # Ulrich Schimke (ulrschimke(AT)aol.com) MATHEMATICA terms = 36; (* F = G001678 *) F[_] = 0; Do[F[x_] = (x^2/(1 + x))*Exp[Sum[ F[x^k]/(k*x^k), {k, 1, j}]] + O[x]^j // Normal, {j, 1, terms + 1}]; G[x_] = 1 + ((1 + x)/x)*F[x] - (F[x]^2 + F[x^2])/(2*x) + O[x]^terms; CoefficientList[G[x] - 1, x] (* Jean-François Alcover, May 25 2012, updated Jan 12 2018 *) PROG (PARI) {a(n) = local(A); if( n<3, n>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( (1 + x) * A - x * (A^2 + subst(A, x, x^2)) / 2, n))}; /* Michael Somos, Jun 13 2014 */ CROSSREFS Cf. A001679. Cf. A000055 (trees by nodes), A000014 (homeomorphically irreducible trees by nodes), A000669 (homeomorphically irreducible planted trees by leaves), A000081 (rooted trees by nodes). Cf. A246403. Sequence in context: A028408 A226452 A037163 * A001679 A030435 A063886 Adjacent sequences:  A059120 A059121 A059122 * A059124 A059125 A059126 KEYWORD nonn,easy,nice AUTHOR Wolfdieter Lang, Jan 09 2001 STATUS approved

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Last modified October 15 12:25 EDT 2019. Contains 328026 sequences. (Running on oeis4.)