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

1,4

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.

Formula

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

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 *)

Prog

(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

Crossrefs

Cf. A002218, A004115, A095983.

Row sums of triangle A123534.

Sequence in context: A096331 A159497 A177595 * A215835 A006423 A067423

Adjacent sequences: A013919 A013920 A013921 * A013923 A013924 A013925

Keyword

nonn,easy,nice

Author

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

Status

approved