OFFSET
0,5
COMMENTS
Hankel transform is A166446(n+2). - Paul Barry, Mar 29 2011
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jean-Luc Baril and Sergey Kirgizov, Lattice paths with a first return decomposition constrained by the maximal height of a pattern, arXiv:2110.02831 [math.CO], 2021.
Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, Motzkin paths with a restricted first return decomposition, Integers (2019) Vol. 19, A46.
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 21.
John Riordan, Enumeration of plane trees by branches and endpoints, J. Comb. Theory (A) 19, 1975, 214-222.
FORMULA
a(n) = sum(binomial(n-2-j, j)*m{j), j=0..floor(n/2)-1), where m(j)=sum(binomial(n, 2k)*binomial(2k, k)/(k+1), k=0..floor(n/2)) is the Motzkin number A001006(j).
G.f.: g=g(z) satisfies z^2g^2-(1-z+z^2)g+1-z+z^2=0
G.f.: [1,1,2,3,...] has g.f. 1/(1-x-x^2-x^4/(1-x-x^2-x^4/(1-x-x^2-x^4/(1... (continued fraction). - Paul Barry, Jul 16 2009
G.f.: c(x^2/(1-x+x^2)) where c(x) is the g.f. of A000108.
G.f.: g(x)=1/(1-x^2/(1-x+x^2-x^2*g(x)))=1/(1-x^2/(1-x+x^2-x^2/(1-x^2/(1-x+x^2-x^2/(1-... (continued fraction). - Paul Barry, Mar 29 2011
D-finite with recurrence (n+2)*a(n) +(-2*n-1)*a(n-1) +(-n+1)*a(n-2) +(2*n-5)*a(n-3) +3*(-n+4)*a(n-4)=0. - R. J. Mathar, Nov 24 2012
G.f.: (1 - x + x^2 - sqrt(1- 2*x - x^2 + 2*x^3 - 3*x^4))/(2*x^2). - Sergei N. Gladkovskii, Oct 04 2013
a(n) ~ sqrt(26+2*sqrt(13)) * ((1+sqrt(13))/2)^n / (4 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 12 2014
EXAMPLE
a(6)=6 because we have the following six ordered trees with 6 edges and no branches of length 1 (hanging from the root): (i) a path of length 6, (ii) a path of length 2 and a path of length 4, (iii) two paths of length 3, (iv) a path of length 4 and a path of length 2, (v) three paths of length 2 and (vi) a path of length 2 at the end of which two paths of length 2 are hanging.
MATHEMATICA
CoefficientList[Series[(1-x+x^2-Sqrt[1-2*x-x^2+2*x^3-3*x^4])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 04 2003
STATUS
approved