A085614 Number of elementary arches of size n.
1, 3, 16, 105, 768, 6006, 49152, 415701, 3604480, 31870410, 286261248, 2604681690, 23957864448, 222399744300, 2080911654912, 19604537460045, 185813170126848, 1770558814528770, 16951376923852800, 162984598242674670
Offset: 1
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..200
- F. Cazals, Combinatorics of Non-Crossing Configurations, Studies in Automatic Combinatorics, Volume II (1997).
- Karl Dilcher, Armin Straub, andChristophe Vignat, Identities for Bernoulli polynomials related to multiple Tornheim zeta functions, arXiv:1903.11759 [math.NT], 2019. See p. 13.
- Loïc Foissy, Free quadri-algebras and dual quadri-algebras, arXiv preprint arXiv:1504.06056 [math.CO], 2015.
- I. M. Gessel, A short proof of the Deutsch-Sagan congruence for connected non crossing graphs, arXiv preprint arXiv:1403.7656 [math.CO], 2014.
- Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
- Vincent Pilaud, Pebble trees, arXiv:2205.06686 [math.CO], 2022.
- Thomas M. Richardson, The three 'R's and Dual Riordan Arrays, arXiv:1609.01193 [math.CO], 2016.
- M. R. Sepanski, On Divisibility of Convolutions of Central Binomial Coefficients, Electronic Journal of Combinatorics, 21 (1) 2014, #P1.32.
- Jian Zhou, Fat and Thin Emergent Geometries of Hermitian One-Matrix Models, arXiv:1810.03883 [math-ph], 2018.
Programs
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Maple
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); -
Mathematica
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 *) Rest[CoefficientList[InverseSeries[Series[x - 3*x^2 + 2*x^3, {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Aug 22 2017 *) (* From Dixon J. Jones, Apr 15 2021: (Start) *) Table[4^n Gamma[(3n + 2)/2]/(Gamma[(n + 2)/2](n + 1)!), {n, 0, 20}] Table[4^n Pochhammer[(n + 2)/2, n]/(n + 1)!, {n, 0, 20}] (* End *) -
PARI
a(n)=if(n<1,0,polcoeff(serreverse(x-3*x^2+2*x^3+x*O(x^n)),n))
Formula
G.f. is the series reversion of x-3*x^2+2*x^3.
a(n) = 2^n*(3*n)!!/((n+1)!*n!!). - Maxim Krikun (krikun(AT)iecn.u-nancy.fr), May 25 2007
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
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
a(n) ~ 2^(n - 3/2) * 3^(3*n/2 - 1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2017
From Dixon J. Jones, Apr 15 2021: (Start)
a(n) = 4^n Gamma((3*n + 2)/2)/(Gamma((n + 2)/2)*(n + 1)!).
a(n) = (4^n*((n + 2)/2)_n)/(n + 1)!, where (x)_k is the Pochhammer symbol. (End)