login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A178834 a(n) counts anti-chains of size two in "0,1,2" Motzkin trees on n edges. 5
0, 0, 1, 5, 23, 91, 341, 1221, 4249, 14465, 48442, 160134, 523872, 1699252, 5472713, 17520217, 55801733, 176942269, 558906164, 1759436704, 5522119250, 17285351782, 53977433618, 168194390290, 523076690018, 1623869984706 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

"0,1,2" trees are rooted trees where each vertex has outdegree zero, one or two. They are counted by the Motzkin numbers.

From Petros Hadjicostas, Jun 02 2020: (Start)

Let A(r,n) be the number of ordered pairs (T, s), where T is a "0,1,2" tree (Motzkin tree) with n edges and s is an r-element anti-chain in T. See Definition 42, p. 30, in Salaam (2008) but we use different notation here.

An r-element anti-chain in a tree is a set of r nodes such that, for every two nodes u and v in the set, u is neither an ancestor nor a descendant of v.

For the current sequence, a(n) = A(r=2, n) for n >= 0.

Let A[r](z) = Sum_{n >= 0} A(r,n)*z^n be the g.f. of the sequence (A(r,n): n >= 0) for fixed r >= 1.

In Theorem 44 (p. 33), Salaam proved that A[r](z) = c_{r-1} * z^(2*r-2) * L(z)^(r-1) * V(z)^r, where c_r = (1/(r + 1))*binomial(2*r, r) is the r-th Catalan number in A000108, L(z) = T(z) = 1/sqrt(1 - 2*z - 3*z^2) is the g.f. of the central trinomial numbers A002426, and V(z) = T(z)*M(z), where M(z) = (1 - z - sqrt(1 - 2*z - 3*z^2))/(2*z^2) is the g.f. of the Motzkin numbers A001006.

It follows (see Table 2.4, p. 39) that A[r](z) = c_{r-1} * z^(2*r-2) * T(z)^(2*r-1) * M(z)^r for fixed r >= 1.

For r = 1, A[r=1](z) = Sum_{n >= 0} A(r=1, n)*z^n = T(z)*M(z) = V(z) is the g.f. of the total number of vertices in all "0,1,2" trees with n edges (i.e., the g.f. of the sequence (A005717(n+1): n >= 0)).

For r = 2, A[r=2](z) = z^2 * T(z)^3 * M(z)^2 is the g.f. of the current sequence. (End)

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

G.f.: z^2 * M(z)^2 * T(z)^3, where M(z) = (1 - z - sqrt(1 - 2*z - 3*z^2))/(2*z^2) is the g.f. of the Motzkin numbers and T(z) = 1/sqrt(1 - 2*z - 3*z^2) is the g.f. of the central trinomial numbers.

Conjecture D-finite with recurrence: -(n-2)*(n+2)*a(n) + (4*n^2-n-8)*a(n-1) + (2*n^2-n-12)*a(n-2) - 3*n*(4*n-3)*a(n-3) - 9*n*(n-1)*a(n-4) = 0. - R. J. Mathar, Jun 14 2016

EXAMPLE

For n = 3, we have a(3) = 5 because there are 5 two-element anti-chains on "0,1,2" Motzkin trees on 3 edges.

MATHEMATICA

M:= (1-z -Sqrt[1-2*z-3*z^2])/(2*z^2); T:= 1/Sqrt[1-2*z-3*z^2]; CoefficientList[Series[z^2*M^2*T^3, {z, 0, 30}], z] (* G. C. Greubel, Jan 21 2019 *)

PROG

(PARI) z='z+O('z^33); M=(1-z-sqrt(1-2*z-3*z^2))/(2*z^2); T=1/sqrt(1-2*z-3*z^2); v=Vec(z^2*M^2*T^3+'tmp); v[1]=0; v

(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); [0, 0] cat Coefficients(R!( (1-x-Sqrt(1-2*x-3*x^2))^2/(4*x^2*Sqrt(1-2*x-3*x^2)^3) )); // G. C. Greubel, Jan 21 2019

(SageMath) ((1-x-sqrt(1-2*x-3*x^2))^2/(4*x^2*sqrt(1-2*x-3*x^2)^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 21 2019

CROSSREFS

Cf. A000108, A002426, A001006, A005717.

Sequence in context: A121329 A246175 A283224 * A331720 A327973 A255457

Adjacent sequences:  A178831 A178832 A178833 * A178835 A178836 A178837

KEYWORD

nonn

AUTHOR

Lifoma Salaam, Dec 27 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 11 22:58 EDT 2022. Contains 356067 sequences. (Running on oeis4.)