OFFSET
0,5
COMMENTS
Essentially the same as A001679. - Eric W. Weisstein, Mar 25 2022
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 = 0.421301852869924921096502830935802411658488216342994235732491571594804013... - 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
KEYWORD
nonn,easy,nice
AUTHOR
Wolfdieter Lang, Jan 09 2001
STATUS
approved