%I #71 May 16 2022 02:07:49
%S 1,3,16,105,768,6006,49152,415701,3604480,31870410,286261248,
%T 2604681690,23957864448,222399744300,2080911654912,19604537460045,
%U 185813170126848,1770558814528770,16951376923852800,162984598242674670
%N Number of elementary arches of size n.
%H Vincenzo Librandi, <a href="/A085614/b085614.txt">Table of n, a(n) for n = 1..200</a>
%H F. Cazals, <a href="http://algo.inria.fr/libraries/autocomb/NCC-html/NCC.html">Combinatorics of Non-Crossing Configurations</a>, Studies in Automatic Combinatorics, Volume II (1997).
%H Karl Dilcher, Armin Straub, andChristophe Vignat, <a href="https://arxiv.org/abs/1903.11759">Identities for Bernoulli polynomials related to multiple Tornheim zeta functions</a>, arXiv:1903.11759 [math.NT], 2019. See p. 13.
%H Loïc Foissy, <a href="http://arxiv.org/abs/1504.06056">Free quadri-algebras and dual quadri-algebras</a>, arXiv preprint arXiv:1504.06056 [math.CO], 2015.
%H I. M. Gessel, <a href="http://arxiv.org/abs/1403.7656">A short proof of the Deutsch-Sagan congruence for connected non crossing graphs</a>, arXiv preprint arXiv:1403.7656 [math.CO], 2014.
%H Elżbieta Liszewska and Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.
%H Vincent Pilaud, <a href="https://arxiv.org/abs/2205.06686">Pebble trees</a>, arXiv:2205.06686 [math.CO], 2022.
%H Thomas M. Richardson, <a href="http://arxiv.org/abs/1609.01193">The three 'R's and Dual Riordan Arrays</a>, arXiv:1609.01193 [math.CO], 2016.
%H M. R. Sepanski, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i1p32">On Divisibility of Convolutions of Central Binomial Coefficients</a>, Electronic Journal of Combinatorics, 21 (1) 2014, #P1.32.
%H Jian Zhou, <a href="https://arxiv.org/abs/1810.03883">Fat and Thin Emergent Geometries of Hermitian One-Matrix Models</a>, arXiv:1810.03883 [math-ph], 2018.
%F G.f. is the series reversion of x-3*x^2+2*x^3.
%F a(n) = 2^n*(3*n)!!/((n+1)!*n!!). - Maxim Krikun (krikun(AT)iecn.u-nancy.fr), May 25 2007
%F G.f.: 1/6*sqrt(3)*sin(1/3*arcsin(6*sqrt(3)*x))-1/2*cos(1/3*arcsin(6*sqrt(3)*x)). - _Vaclav Kotesovec_, Oct 21 2012
%F Conjecture: n*(n-1)*a(n) +(n-1)*(n-2)*a(n-1) -12*(3*n-5)*(3*n-7)*a(n-2) -12*(3*n-8)*(3*n-10)*a(n-3) = 0. - _R. J. Mathar_, Oct 18 2013
%F a(n) ~ 2^(n - 3/2) * 3^(3*n/2 - 1) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Aug 22 2017
%F From _Dixon J. Jones_, Apr 15 2021: (Start)
%F a(n) = A206300(n)/2 = abs(A224884(n))/2 for n>=1.
%F a(n) = 4^n Gamma((3*n + 2)/2)/(Gamma((n + 2)/2)*(n + 1)!).
%F a(n) = (4^n*((n + 2)/2)_n)/(n + 1)!, where (x)_k is the Pochhammer symbol. (End)
%p with(combstruct); ar := {EA = Union(Sequence(EA, card >= 2), Prod(Z, Sequence(EA), Sequence(EA))), C=Union(Z, Prod(Z,Z,Sequence(EA), Sequence(EA), Sequence(Union(Sequence(EA,card>=1), Prod(Z,Sequence(EA),Sequence(EA))))))}; seq(count([EA,ar], size=i),i=1..20);
%t Rest[CoefficientList[Series[1/6*Sqrt[3]*Sin[1/3*ArcSin[6*Sqrt[3]*x]] - 1/2*Cos[1/3*ArcSin[6*Sqrt[3]*x]],{x,0,20}],x]] (* _Vaclav Kotesovec_, Oct 21 2012 *)
%t Rest[CoefficientList[InverseSeries[Series[x - 3*x^2 + 2*x^3, {x, 0, 20}], x],x]] (* _Vaclav Kotesovec_, Aug 22 2017 *)
%t (* From _Dixon J. Jones_, Apr 15 2021: (Start) *)
%t Table[4^n Gamma[(3n + 2)/2]/(Gamma[(n + 2)/2](n + 1)!), {n, 0, 20}]
%t Table[4^n Pochhammer[(n + 2)/2, n]/(n + 1)!, {n, 0, 20}] (* End *)
%o (PARI) a(n)=if(n<1,0,polcoeff(serreverse(x-3*x^2+2*x^3+x*O(x^n)),n))
%Y Cf. A143018, A206300.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Jul 10 2003