OFFSET
6,1
COMMENTS
"0,1,2" trees are rooted trees where each vertex has outdegree zero, one, or two. They are counted by the Motzkin numbers A001006.
LINKS
Martin Klazar, Twelve countings with rooted plane trees, European Journal of Combinatorics, 18(2) (1997), 195-210. [The author counts anti-chains for some kinds of rooted trees but not for Motzkin rooted trees.]
Lifoma Salaam, Combinatorial statistics on phylogenetic trees, Ph.D. Dissertation, Howard University, Washington D.C., 2008; see Definition 42 (p. 30), Theorem 44 (p. 33), and Table 2.4 (p. 39).
FORMULA
EXAMPLE
For n=6, we list below all a(6) = 5 four-element anti-chains in Motzkin rooted trees with 6 edges:
A A A
/ \ / \ / \
/ \ / \ / \
B C B C B C
/ \ / \ / \ / \
/ \ / \ / \ / \
D E F G D E D E
{D, E, F, G} / \ / \
/ \ / \
F G F G
{C, D, F, G} {C, E, F, G}
A A
/ \ / \
/ \ / \
B C B C
/ \ / \
/ \ / \
D E D E
/ \ / \
/ \ / \
F G F G
{B, E, F, G} {B, D, F, G}
PROG
(PARI) default(seriesprecision, 50);
M(z) = (1 - z - sqrt(1 - 2*z - 3*z^2))/(2*z^2);
T(z) = 1/sqrt(1 - 2*z - 3*z^2);
for(n=0, 30, print1(polcoef(5*z^6*T(z)^7*M(z)^4, n, z), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Petros Hadjicostas, Jun 03 2020
STATUS
approved