This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A054727 #56 Aug 23 2020 06:32:57
%S A054727 1,2,7,33,181,1083,6854,45111,305629,2117283,14929212,106790500,
%T A054727 773035602,5652275723,41683912721,309691336359,2315772552485,
%U A054727 17415395593371,131632335068744,999423449413828,7618960581522348,58295017292748756,447517868947619432,3445923223190363608
%N A054727 Number of forests of rooted trees with n nodes on a circle without crossing edges.
%H A054727 Andrew Howroyd, <a href="/A054727/b054727.txt">Table of n, a(n) for n = 1..200</a>
%H A054727 C. Banderier and D. Merlini, <a href="http://algo.inria.fr/banderier/Papers/infjumps.ps">Lattice paths with an infinite set of jumps</a>, FPSAC02, Melbourne, 2002.
%H A054727 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 A054727 Guillermo Esteban, Clemens Huemer, Rodrigo I. Silveira, <a href="https://arxiv.org/abs/2003.00524">New production matrices for geometric graphs</a>, arXiv:2003.00524 [math.CO], 2020.
%H A054727 Philippe Flajolet, <a href="http://algo.inria.fr/flajolet/Publications">Source</a>
%H A054727 Philippe Flajolet, <a href="http://algo.inria.fr/libraries/autocomb/nc-configs-html/nc-configs1.html">Enumeration of planar configurations in computational geometry</a>
%H A054727 P. Flajolet and M. Noy, <a href="https://dx.doi.org/10.1016/S0012-365X(98)00372-0">Analytic combinatorics of noncrossing configurations</a>, Discrete Math. 204 (1999), 203-229.
%H A054727 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 486, 502
%H A054727 Samuele Giraudo, <a href="https://arxiv.org/abs/1709.08416">Combalgebraic structures on decorated cliques</a>, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 8, arXiv:1709.08416 [math.CO], 2017.
%H A054727 Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, <a href="https://arxiv.org/abs/1706.03357">Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing</a>, arXiv:1706.03357 [cs.CL], 2017.
%F A054727 a(n) = Sum_{j=1..n} binomial(n, j-1) * binomial(3*n-2*j-1, n-j) / (2*n - j).
%F A054727 G.f. A(x) satisfies 2*A(x)^2=x*(1-sqrt(1-4*A(x)))*(1-A(x)). - _Vladimir Kruchinin_, Nov 25 2011
%F A054727 From _Peter Bala_, Nov 07 2015: (Start)
%F A054727 O.g.f. A(x) = revert(x/((1 + x)*C(x))), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f for the Catalan numbers A000108.
%F A054727 Row sums of A094021. (End)
%F A054727 Conjecture: -2(37*n-80) *(n-1) *(2*n-1) *a(n) +2*(592*n^3-3056*n^2+5045*n-2665) *a(n-1) +2*(148*n^3-986*n^2+2021*n-1255) *a(n-2) -5*(n-5) *(n-2) *(37*n-43) *a(n-3)=0. - _R. J. Mathar_, Apr 30 2018
%F A054727 a(n) ~ c * d^n / (sqrt(Pi) * n^(3/2)), where d = 8.2246915409778560686084627753... is the real root of the equation 5 - 8*d - 32*d^2 + 4*d^3 = 0 and c = 0.07465927842190452347018812862935237... is the positive real root of the equation -125 + 22376*c^2 + 8880*c^4 + 592*c^6 = 0. - _Vaclav Kotesovec_, Apr 30 2018
%p A054727 ZZ:=[F,{F=Union(Epsilon,ZB),ZB=Prod(Z1,P),P=Sequence(B),B=Prod(P,Z1,P),Z1=Prod(Z,F)}, unlabeled]: seq(count(ZZ,size=n),n=1..20); # _Zerinvary Lajos_, Apr 22 2007
%t A054727 a[n_] := (3*n-3)!/((n-1)!*(2*n-1)!)*HypergeometricPFQ[{1-2*n, 1-n, -n}, {3/2 - 3*n/2, 2 - 3*n/2}, -1/4]; Table[a[n], {n, 1, 20}] (* _Jean-François Alcover_, Sep 05 2012, after formula *)
%o A054727 (PARI) N=33; x='x+O('x^N); Vec(serreverse(x/((1+x)*(1-sqrt(1-4*x))/(2*x)))) \\ _Joerg Arndt_, May 25 2016
%Y A054727 Row sums of A094021.
%Y A054727 Cf. A006013, A000108.
%K A054727 nonn,easy
%O A054727 1,2
%A A054727 _Philippe Flajolet_, Apr 20 2000