0,2

The cited preprint gives an asymptotic estimate for the number of trees as the path length goes to infinity, for t-ary trees, t >= 2. This sequence corresponds to t=2.

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..200 from Vincenzo Librandi)

G. Seroussi, On the number of t-ary trees with a given path length, arXiv:cs/0509046 [cs.DM], 2005-2007; Algorithmica 46(3), 557-565, 2006.

G.f.: B(w, 1) - 1, where B(w, z) satisfies the functional equation B(w, z) = z B(w, wz)^2 + 1. B(w, z) is the g.f. for the number of binary trees of given path length and number of nodes (see Knuth Vol. 1 Sec. 2.3.4.5); B(1, z) is the g.f. for the Catalan numbers; for B(w, w) see A108643.

a(1) = 2 because there are two binary trees of path length 1: a root with a left child and a root with a right child.

a(2) = 1 because there is just one binary tree of path length 2: a root with its two children.

terms = 44; B[_, _] = 0;

Do[B[w_, z_] = Series[z B[w, w z]^2 + 1, {w, 0, terms-1}, {z, 0, terms-1}] // Normal, {terms-1}];

CoefficientList[B[w, 1] - 1, w] (* Jean-François Alcover, Dec 03 2018 *)

Cf. A106182.

Sequence in context: A379200 A375049 A129159 * A193915 A101621 A086484

Adjacent sequences: A095827 A095828 A095829 * A095831 A095832 A095833

Gadiel Seroussi (seroussi(AT)hpl.hp.com), Jul 10 2004

approved