This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A335355 #25 Jun 12 2022 03:00:19
%S A335355 5,55,420,2600,14175,70665,329800,1462680,6228945,25661875,102847560,
%T A335355 402706500,1545715325,5831511195,21671504880,79475234200,288043346370,
%U A335355 1033030388790,3669961024940,12927078062500,45182780785500,156811313843420,540722493900480,1853503409060160
%N A335355 a(n) counts anti-chains of size four in "0,1,2" Motzkin trees on n edges.
%C A335355 "0,1,2" trees are rooted trees where each vertex has outdegree zero, one, or two. They are counted by the Motzkin numbers A001006.
%C A335355 A005717(n+1) is the total number of vertices (= anti-chains of size 1) in all "0,1,2" trees with n edges, A178834(n) is the total number of anti-chains of size 2 in all "0,1,2" trees on n edges, and A335349(n) is the total number of anti-chains of size 3 in all "0,1,2" trees on n edges.
%C A335355 It would be interesting to examine whether there is an interpretation of this sequence and sequences A178834 and A335349 in terms of Motzkin paths. (Salaam (2008) worked with different families of rooted trees, but not with Motzkin paths.)
%H A335355 Martin Klazar, <a href="https://doi.org/10.1006/eujc.1995.0095">Twelve countings with rooted plane trees</a>, 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.]
%H A335355 Lifoma Salaam, <a href="https://search.proquest.com/docview/193997569">Combinatorial statistics on phylogenetic trees</a>, Ph.D. Dissertation, Howard University, Washington D.C., 2008; see Definition 42 (p. 30), Theorem 44 (p. 33), and Table 2.4 (p. 39).
%F A335355 G.f.: A000108(r-1) * z^(2*r-2) * T(z)^(2*r-1) * M(z)^r = 5 * z^6 * T(z)^7 * M(z)^4 (with r = 4), where M(z) = (1 - z - sqrt(1 - 2*z - 3*z^2)) / (2*z^2) is the g.f. of the Motzkin numbers A001006 and T(z) = 1 / sqrt(1 - 2*z - 3*z^2) is the g.f. of the central trinomial numbers A002426.
%e A335355 For n=6, we list below all a(6) = 5 four-element anti-chains in Motzkin rooted trees with 6 edges:
%e A335355 A A A
%e A335355 / \ / \ / \
%e A335355 / \ / \ / \
%e A335355 B C B C B C
%e A335355 / \ / \ / \ / \
%e A335355 / \ / \ / \ / \
%e A335355 D E F G D E D E
%e A335355 {D, E, F, G} / \ / \
%e A335355 / \ / \
%e A335355 F G F G
%e A335355 {C, D, F, G} {C, E, F, G}
%e A335355 A A
%e A335355 / \ / \
%e A335355 / \ / \
%e A335355 B C B C
%e A335355 / \ / \
%e A335355 / \ / \
%e A335355 D E D E
%e A335355 / \ / \
%e A335355 / \ / \
%e A335355 F G F G
%e A335355 {B, E, F, G} {B, D, F, G}
%o A335355 (PARI) default(seriesprecision, 50);
%o A335355 M(z) = (1 - z - sqrt(1 - 2*z - 3*z^2))/(2*z^2);
%o A335355 T(z) = 1/sqrt(1 - 2*z - 3*z^2);
%o A335355 for(n=0, 30, print1(polcoef(5*z^6*T(z)^7*M(z)^4, n, z), ", "))
%Y A335355 Cf. A000108, A001006, A002426, A005717, A178834, A335349.
%K A335355 nonn
%O A335355 6,1
%A A335355 _Petros Hadjicostas_, Jun 03 2020