A007297 Number of connected graphs on n labeled nodes on a circle with straight-line edges that don't cross.

1, 1, 4, 23, 156, 1162, 9192, 75819, 644908, 5616182, 49826712, 448771622, 4092553752, 37714212564, 350658882768, 3285490743987, 30989950019532, 294031964658430, 2804331954047160, 26870823304476690, 258548658860327880

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• easy
• nice

Comments

Apart from the initial 1, reversion of g.f. for A162395 (squares with signs): see A263843.

Examples

			G.f. = x*(1 + x + 4*x^2 + 23*x^3 + 156*x^4 + 1162*x^5 + 9192*x^6 + 75819*x^7 + ...).
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..999 (first 101 terms from T. D. Noe)
• M. Aguiar and J.-L. Loday, Quadri-algebras, J. Pure Appl. Algebra, 191 (2004), 205-221.
• R. Bacher, On generating series of complementary plane trees arXiv:math/0409050 [math.CO], 2004.
• P. Barry and A. Hennessy, Four-term Recurrences, Orthogonal Polynomials and Riordan Arrays, Journal of Integer Sequences, 2012, article 12.4.2. - From _N. J. A. Sloane_, Sep 21 2012
• F. R. Bernhart & N. J. A. Sloane, Emails, April-May 1994
• F. Cazals, Combinatorics of Non-Crossing Configurations, Studies in Automatic Combinatorics, Volume II (1997).
• E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004; J. Num. Theory 117 (2006), 191-215.
• C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358 (column sums in Table 2).
• Michael Drmota, Anna de Mier, Marc Noy, Extremal statistics on non-crossing configurations, Discrete Math. 327 (2014), 103--117. MR3192420. See Eq. (3). - N. J. A. Sloane, May 18 2014
• Guillermo Esteban, Clemens Huemer, Rodrigo I. Silveira, New production matrices for geometric graphs, arXiv:2003.00524 [math.CO], 2020.
• Sen-Peng Eu, Shu-Chung Liu and Yeong-Nan Yeh, On the congruences of some combinatorial numbers, Stud. Appl. Math. vol. 116 (2006) pp. 135-144.
• P. Flajolet and M. Noy, Analytic Combinatorics of Non-crossing Configurations
• P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
• P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 486.
• Loïc Foissy, Free quadri-algebras and dual quadri-algebras, 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.
• Marco Kuhlmann, Tabulation of Noncrossing Acyclic Digraphs, arXiv:1504.04993 [cs.DS], 2015.
• M. Kuhlmann, P. Jonsson, Parsing to Noncrossing Dependency Graphs, Transactions of the Association for Computational Linguistics, vol. 3, pp. 559-570, 2015.
• Sara Madariaga, Gröbner-Shirshov bases for the non-symmetric operads of dendriform algebras and quadri-algebras, arXiv:1304.5184 [math.RA], 2013.
• B. Vallette, Manin products, Koszul duality, Loday algebras and Deligne conjecture, arXiv:math/0609002 [math.QA], 2006-2007; J. Reine Angew. Math. 620 (2008), 105-164.
• Gus Wiseman, The a(5) = 156 connected non-crossing graphs.
• Gus Wiseman, Constructive Mathematica program for A007297.
• Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing, arXiv:1706.03357 [cs.CL], 2017.
• Index entries for reversions of series

Crossrefs

Cf. A162395, A000290. 4th row of A107111. Row sums of A089434.

See A263843 for a variant.

Cf. A000108 (non-crossing set partitions), A001006, A001187, A054726 (non-crossing graphs), A054921, A099947, A194560, A293510, A323818, A324167, A324169, A324173.

Programs

Maple

A007297:=proc(n) if n = 1 then 1 else add(binomial(3*n - 3, n + j)*binomial(j - 1, j - n + 1), j = n - 1 .. 2*n - 3)/(n - 1); fi; end;

Mathematica

CoefficientList[ InverseSeries[ Series[(x-x^2)/(1+x)^3, {x, 0, 20}], x], x] // Rest (* From Jean-François Alcover, May 19 2011, after PARI prog. *)
Table[Binomial[3n, 2n+1] Hypergeometric2F1[1-n, n, 2n+2, -1]/n, {n, 1, 20}] (* Vladimir Reshetnikov, Oct 25 2015 *)

PARI

a(n)=if(n<0,0,polcoeff(serreverse((x-x^2)/(1+x)^3+O(x^(n+2))),n+1)) /* Ralf Stephan */

Formula

Apart from initial term, g.f. is the series reversion of (x-x^2)/(1+x)^3 (A162395). See A263843. - Vladimir Kruchinin, Feb 08 2013

G.f.: (g-z)/z, where g=-1/3+(2/3)*sqrt(1+9z)*sin((1/3)*arcsin((2+27z+54z^2)/2/(1+9*z)^(3/2))). - Emeric Deutsch, Dec 02 2002

a(n) = (1/n)*Sum_{k=0..n} binomial(3n, n-k-1)*binomial(n+k-1, k). - Paul Barry, May 11 2005

a(n) = 4^(n-1)*(Gamma(3*n/2-1)/Gamma(n/2+1)/Gamma(n) -Gamma((3*n-1)/2)/ Gamma( (n+1)/2)/Gamma(n+1)). - Mark van Hoeij, Aug 27 2005, adapted to offset Feb 21 2020 by R. J. Mathar

a(n) = 4^n * binomial(3*n/2, n/2) / (9*n-6) - 4^(n-1) * binomial(3*(n-1)/2, (n-1)/2 ) / n. - Mark van Hoeij, Aug 27 2005, adapted to offset Feb 21 2020 by R. J. Mathar

D-finite with recurrence: n*(n-1)*(3*n-4)*a(n) +36*(n-1)*a(n-1) -12*(3*n-8)*(3*n-1)*(3*n-7)*a(n-2)=0. - Mark van Hoeij, Aug 27 2005, adapted to offset Feb 21 2020 by R. J. Mathar

a(n) = (1/n)*Sum_{k=0..n} C(3n, k)*C(2n-k-2, n-1). - Paul Barry, Sep 27 2005

a(n) ~ (2-sqrt(3)) * 6^n * 3^(n/2) / (sqrt(2*Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 17 2014

a(n) = binomial(3*n,2*n+1)*hypergeom([1-n,n], [2*n+2], -1)/n. - Vladimir Reshetnikov, Oct 25 2015

a(n) = 2*A078531(n) - A085614(n+1). - Vladimir Reshetnikov, Apr 24 2016

Extensions

Better description from Philippe Flajolet, Apr 20 2000

More terms from James Sellers, Aug 21 2000

Definition revised and initial a(1)=1 added by N. J. A. Sloane, Nov 05 2015 at the suggestion of Axel Boldt. Some of the formulas may now need to be adjusted slightly.