A013922 Number of labeled connected graphs with n nodes and 0 cutpoints (blocks or nonseparable graphs).

0, 1, 1, 10, 238, 11368, 1014888, 166537616, 50680432112, 29107809374336, 32093527159296128, 68846607723033232640, 290126947098532533378816, 2417684612523425600721132544, 40013522702538780900803893881856

Offset: 1

Author

Stanley Selkow (sms(AT)owl.WPI.EDU)

Keywords

• nonn
• easy
• nice

Comments

Or, number of labeled 2-connected graphs with n nodes.

References

• Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p.402.
• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 9.
• R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.20(b), g(n).

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..50 (terms 1..25 from R. W. Robinson)
• Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002.
• Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002 [Local copy, with permission]
• Thomas Lange, Biconnected reliability, Hochschule Mittweida (FH), Fakultät Mathematik/Naturwissenschaften/Informatik, Master's Thesis, 2015.
• Andrés Santos, Density Expansion of the Equation of State, in A Concise Course on the Theory of Classical Liquids, Volume 923 of the series Lecture Notes in Physics, pp 33-96, 2016. DOI:10.1007/978-3-319-29668-5_3. See Reference 40.
• S. Selkow, The enumeration of labeled graphs by number of cutpoints, Discr. Math. 185 (1998), 183-191.

Crossrefs

Cf. A002218, A004115, A095983.

Row sums of triangle A123534.

Programs

Mathematica

seq[n_] := CoefficientList[Log[x/InverseSeries[x*D[Log[Sum[2^Binomial[k, 2]*x^k/k!, {k, 0, n}] + O[x]^n], x]]], x]*Range[0, n-2]!;
seq[16] (* Jean-François Alcover, Aug 19 2019, after Andrew Howroyd *)

PARI

seq(n)={Vec(serlaplace(log(x/serreverse(x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n)))))), -n)} \\ Andrew Howroyd, Sep 26 2018

Formula

Harary and Palmer give e.g.f. in Eqn. (1.3.3) on page 10.