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%I #51 Mar 03 2024 14:50:30
%S 1,0,1,1,2,3,6,11,22,43,87,176,362,748,1560,3270,6897,14613,31104,
%T 66459,142518,306603,661572,1431363,3104619,6749390,14704387,32098729,
%U 70198656,153785705,337443785,741551614,1631910081,3596083215
%N Number of ordered trees with n edges and having no branches of length 1.
%C Hankel transform is A166446(n+2). - _Paul Barry_, Mar 29 2011
%H Vincenzo Librandi, <a href="/A026418/b026418.txt">Table of n, a(n) for n = 0..1000</a>
%H Jean-Luc Baril and Sergey Kirgizov, <a href="https://arxiv.org/abs/2110.02831">Lattice paths with a first return decomposition constrained by the maximal height of a pattern</a>, arXiv:2110.02831 [math.CO], 2021.
%H Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, <a href="http://math.colgate.edu/~integers/t46/t46.Abstract.html">Motzkin paths with a restricted first return decomposition</a>, Integers (2019) Vol. 19, A46.
%H Jean-Luc Baril and José Luis Ramírez, <a href="https://arxiv.org/abs/2302.12741">Descent distribution on Catalan words avoiding ordered pairs of Relations</a>, arXiv:2302.12741 [math.CO], 2023.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry3/barry93.html">Continued fractions and transformations of integer sequences</a>, JIS 12 (2009) 09.7.6.
%H Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2402.16111">Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis</a>, arXiv:2402.16111 [math.CO], 2024. See p. 21.
%H John Riordan, <a href="http://dx.doi.org/10.1016/S0097-3165(75)80010-0">Enumeration of plane trees by branches and endpoints</a>, J. Comb. Theory (A) 19, 1975, 214-222.
%F 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).
%F G.f.: g=g(z) satisfies z^2g^2-(1-z+z^2)g+1-z+z^2=0
%F 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
%F G.f.: c(x^2/(1-x+x^2)) where c(x) is the g.f. of A000108.
%F 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
%F 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
%F 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
%F a(n) ~ sqrt(26+2*sqrt(13)) * ((1+sqrt(13))/2)^n / (4 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 12 2014
%e 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.
%t 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 *)
%Y Cf. A001006.
%K nonn
%O 0,5
%A _Emeric Deutsch_, Dec 04 2003