

A095830


Number of binary trees of path length n.


6



1, 2, 1, 4, 4, 2, 14, 8, 12, 28, 21, 52, 52, 72, 92, 160, 212, 178, 446, 360, 628, 920, 918, 1568, 1784, 2676, 2960, 4724, 5360, 7280, 10876, 10936, 17484, 21732, 28469, 34224, 48648, 61232, 78196, 105680, 120904, 178848, 217404, 279312
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OFFSET

0,2


COMMENTS

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


LINKS

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 tary trees with a given path length, arXiv:cs/0509046 [cs.DM], 20052007; Algorithmica 46(3), 557565, 2006.


FORMULA

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.


EXAMPLE

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.


MATHEMATICA

terms = 44; B[_, _] = 0;
Do[B[w_, z_] = Series[z B[w, w z]^2 + 1, {w, 0, terms1}, {z, 0, terms1}] // Normal, {terms1}];
CoefficientList[B[w, 1]  1, w] (* JeanFrançois Alcover, Dec 03 2018 *)


CROSSREFS

Cf. A106182.
Sequence in context: A051289 A090802 A129159 * A193915 A101621 A086484
Adjacent sequences: A095827 A095828 A095829 * A095831 A095832 A095833


KEYWORD

nonn


AUTHOR

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


STATUS

approved



