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 A001679 Number of series-reduced rooted trees with n nodes. (Formerly M0327 N0123) 5
 1, 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, 3577169927 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Also known as homeomorphically irreducible rooted trees, or rooted trees without nodes of degree 2. Prime values begin a(4) = a(5) = 2, a(11) = 71, a(12) = 137, a(18) = 7841, a(19) = 15749, a(29) = 20665781. Semiprime values begin a(6) = 4 = 2^2, a(7) = 6 = 2 * 3, a(10) = 39 = 3 * 13, a(14) = 511 = 7 * 73, a(15) = 995 = 5 * 199, a(20) = 31835 = 5 * 6367, a(26) = 2330381 = 1307 * 1783, a(32) = 186447559 = 2437 * 76507, a(36) = 3577169927 = 41 * 87248047. - Jonathan Vos Post, Jun 18 2005 REFERENCES Cameron, Peter J. Some treelike objects. Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155-183. MR0891613 (89a:05009). See p. 155. - N. J. A. Sloane, Apr 18 2014 D. G. Cantor, personal communication. F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 62, Eq. (3.3.9). 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). LINKS N. J. A. Sloane, Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 N. J. A. Sloane, Illustration of initial terms P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183. F. Harary, G. Prins, The number of homeomorphically irreducible trees and other species, Acta Math. 101 (1959) 141-162, W(x,y) equation (9a) Eric Weisstein's World of Mathematics, Series-Reduced Tree. FORMULA G.f. = 1 + ((1+x)*f(x) - (f(x)^2+f(x^2))/2)/x where f(x) is g.f. for A001678 (homeomorphically irreducible planted trees by nodes). 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. = 1 + 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: G001679 := G0temp / x + G0temp - (G0temp^2+eval(G0temp, x=x^2))/(2*x): A001679 := 0, seq(coeff(G001679, x^i), i=1..n); # Ulrich Schimke (ulrschimke(AT)aol.com) MATHEMATICA terms = 37; (* 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], x] (* Jean-François Alcover, 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))}; CROSSREFS Apart from initial term, same as A059123. 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: A226452 A037163 A059123 * A030435 A063886 A003000 Adjacent sequences:  A001676 A001677 A001678 * A001680 A001681 A001682 KEYWORD nonn AUTHOR EXTENSIONS Additional comments from Michael Somos, Oct 10 2003 Maple program adapted for Maple 16 or higher version by Vaclav Kotesovec, Jun 26 2014 STATUS approved

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Last modified March 21 22:19 EDT 2019. Contains 321382 sequences. (Running on oeis4.)