|
%I #39 Dec 27 2020 19:25:53
%S 0,1,1,10,238,11368,1014888,166537616,50680432112,29107809374336,
%T 32093527159296128,68846607723033232640,290126947098532533378816,
%U 2417684612523425600721132544,40013522702538780900803893881856
%N Number of labeled connected graphs with n nodes and 0 cutpoints (blocks or nonseparable graphs).
%C Or, number of labeled 2-connected graphs with n nodes.
%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p.402.
%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 9.
%D R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.20(b), g(n).
%H Andrew Howroyd, <a href="/A013922/b013922.txt">Table of n, a(n) for n = 1..50</a> (terms 1..25 from R. W. Robinson)
%H Huantian Cao, <a href="http://cobweb.cs.uga.edu/~rwr/STUDENTS/hcao.html">AutoGF: An Automated System to Calculate Coefficients of Generating Functions</a>, thesis, 2002.
%H Huantian Cao, <a href="/A000009/a000009.pdf">AutoGF: An Automated System to Calculate Coefficients of Generating Functions</a>, thesis, 2002 [Local copy, with permission]
%H Thomas Lange, <a href="https://monami.hs-mittweida.de/frontdoor/index/index/docId/6733">Biconnected reliability</a>, Hochschule Mittweida (FH), Fakultät Mathematik/Naturwissenschaften/Informatik, Master's Thesis, 2015.
%H Andrés Santos, <a href="https://doi.org/10.1007/978-3-319-29668-5_3">Density Expansion of the Equation of State</a>, 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.
%H S. Selkow, <a href="https://doi.org/10.1016/S0012-365X(97)00170-2">The enumeration of labeled graphs by number of cutpoints</a>, Discr. Math. 185 (1998), 183-191.
%F Harary and Palmer give e.g.f. in Eqn. (1.3.3) on page 10.
%t 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]!;
%t seq[16] (* _Jean-François Alcover_, Aug 19 2019, after _Andrew Howroyd_ *)
%o (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
%Y Cf. A002218, A004115, A095983.
%Y Row sums of triangle A123534.
%K nonn,easy,nice
%O 1,4
%A Stanley Selkow (sms(AT)owl.WPI.EDU)
|
