A097999 Number of 2-connected outerplanar graphs on n labeled nodes.

1, 9, 132, 2700, 70920, 2275560, 86264640, 3772681920, 186972105600, 10355595465600, 633892275878400, 42495895579737600, 3096545573029708800, 243680880958010496000, 20596410256606119936000, 1860881636529774802944000, 178975197401013144907776000, 18256461815785805072068608000

Offset: 3

Author

Steven Finch, Sep 08 2004

Keywords

• nonn

References

• Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 424, see B(x).

Links

• Andrew Howroyd, Table of n, a(n) for n = 3..200
• Bodirsky, M., Giménez, O., Kang, M., & Noy, M., The asymptotic number of outerplanar graphs and series-parallel graphs, in Proceedings of European Conference on Combinatorics, Graph Theory, and Applications (EuroComb05), DMTCS Proceedings Volume AE (pp. 383-388). [Cached copy, with permission]
• M. Bodirsky and M. Kang, The asymptotic number of outerplanar graphs. [Only the abstract has been archived]
• M. Drmota, O. Gimenez, Marc Noy, Vertices of given degree in series-parallel graphs, Random Struct. Algor. 36 (3) (2010), 251-371, Lemma 2.3
• Steven R. Finch, Planar graph growth constants.

Crossrefs

Cf. A097998, A098000.

Programs

Mathematica

offset = 3; terms = 15; egf = (-3 + 2*x - 3*x^2)/16 + (3 - x)*(Sqrt[1 - 6*x + x^2]/16) + Log[(3 - x - Sqrt[1 - 6*x + x^2])/2]/2; Drop[ CoefficientList[ egf + O[x]^(terms + offset), x]*Range[0, terms + offset - 1]!, offset] (* Jean-François Alcover, Nov 05 2016, after Vladeta Jovovic *)

PARI

seq(n)={Vec(serlaplace(intformal((1 + 5*x - sqrt(1 - 6*x + x^2 + O(x^n)))/8 - x)))} \\ Andrew Howroyd, Feb 12 2021

Formula

Recurrence known, see Bodirsky and Kang.

E.g.f.: (-3+2*x-3*x^2)/16+(3-x)*sqrt(1-6*x+x^2)/16+log((3-x-sqrt(1-6*x+x^2))/2)/2. - Vladeta Jovovic, Jun 26 2007

a(n) ~ 2^(-5/2) * sqrt(3*sqrt(2)-4) * (1+sqrt(2))^(2*n-2) * n^(n-2) / exp(n). - Vaclav Kotesovec, Nov 05 2016

Extensions

Terms a(18) and beyond from Andrew Howroyd, Feb 12 2021