0,5

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 (jvospost3(AT)gmail.com), Jun 18 2005

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

D. G. Cantor, personal communication.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 62, Eq. (3.3.9).

F. Harary and G. Prins, The number of homeomorphically irreducible trees and other species, Acta Math., 101 (1959), 141-162.

N. J. A. Sloane, Table of n, a(n) for n = 0..500

N. J. A. Sloane, Illustration of initial terms

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for sequences related to rooted trees

Index entries for sequences related to trees

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).

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); # from UlrSchimke(AT)aol.com

(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))

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).

Sequence in context: A028408 A037163 A059123 this_sequence A030435 A063886 A003000

Adjacent sequences: A001676 A001677 A001678 this_sequence A001680 A001681 A001682

nonn

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from Michael Somos, Oct 10, 2003