cp's OEIS Frontend

A002831 Number of 3-edge-colored connected trivalent graphs with 2n nodes.

%I A002831 M3424 N1388 #83 Feb 16 2025 08:32:27
%S A002831 1,4,11,60,318,2806,29359,396196,6231794,112137138,2249479114,
%T A002831 49691965745,1197158348160,31230408793660,876971159096883,
%U A002831 26374570956403684,845812191249484022,28812214090645864661,1038982259432805270094,39540452134474760212909
%N A002831 Number of 3-edge-colored connected trivalent graphs with 2n nodes.
%C A002831 In a letter to N. J. A. Sloane dated Feb 04 1971 (see link), R. C. Read enclosed a table listing 14 sequences, all of which, he says, appeared in his 1958 Ph.D. thesis. The values he gave for terms a(5) and a(6) in the present sequence are apparently incorrect (the terms given here are correct; the incorrect terms are shown in A246598). - _N. J. A. Sloane_, Sep 08 2014
%C A002831 Comment from _Max Alekseyev_, Sep 09 2014: the relationship between "all graphs" and "connected graphs" is of course a version of the Euler transform - see for example the third formula in the Euler Transform link.
%C A002831 From _Sasha Kolpakov_, Dec 17 2017: (Start)
%C A002831 Number of oriented unrooted pavings (after Arques & Koch, Spehner, Lienhardt) with 2n darts.
%C A002831 Also the number of conjugacy classes of free index 2n subgroups in the free product Z_2*Z_2*Z_2. (End)
%D A002831 R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
%D A002831 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002831 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002831 Andrew Howroyd, <a href="/A002831/b002831.txt">Table of n, a(n) for n = 1..30</a>
%H A002831 Rémi Bottinelli, Laura Ciobanu, and Alexander Kolpakov, <a href="https://doi.org/10.1007/s00229-021-01321-7">Three-dimensional maps and subgroup growth</a>, manuscripta math. (2021).
%H A002831 L. Ciobanu and A. Kolpakov, <a href="https://arxiv.org/abs/1712.01418">Three-dimensional maps and subgroup growth</a>, arXiv:1712.01418 [math.GR], 2017.
%H A002831 R. C. Read, <a href="/A002831/a002831.pdf">Letter to N. J. A. Sloane, Feb 04 1971</a>
%H A002831 Neriman Tokcan, Jonathan Gryak, Kayvan Najarian, and Harm Derksen, <a href="https://arxiv.org/abs/2005.12988">Algebraic Methods for Tensor Data</a>, arXiv:2005.12988 [math.RT], 2020.
%H A002831 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EulerTransform.html">Euler Transform</a>
%F A002831 G.f.: sum(mobius(k) * log(G(x^k)) / k, k >= 1) where G(x) is the g.f. for A002830. - _Sean A. Irvine_, Sep 09 2014
%F A002831 Asymptotics: a(n) ~ (2/Pi)^(1/2)*(2/e)^n*n^{n - 1/2}; cf. Ciobanu and Kolpakov in Links. - _Sasha Kolpakov_, Dec 17 2017
%t A002831 terms = 20;
%t A002831 permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t k; s += t]; s!/m];
%t A002831 b[k_, q_] := If[OddQ[q], If[OddQ[k], 0, j = k/2; q^j (2 j)!/(j! 2^j)], Sum[ Binomial[k, 2 j] q^j (2 j)!/(j! 2^j), {j, 0, Quotient[k, 2]}]];
%t A002831 pm[v_] := Module[{p = Total[x^v]}, Product[ b[Coefficient[p, x, i], i], {i, 1, Exponent[p, x]}]];
%t A002831 a2830[n_] := Module[{s = 0}, Do[ s += permcount[p] pm[p]^3, {p, IntegerPartitions[2 n]}]; s/(2 n)!];
%t A002831 G[x_] = 1 + Sum[a2830[n] x^n, {n, 1, terms+1}];
%t A002831 gf = Sum[MoebiusMu[k] Log[G[x^k]]/k, {k, 1, terms+1}] + O[x]^(terms+1);
%t A002831 CoefficientList[gf, x] // Rest (* _Jean-François Alcover_, Jul 02 2018, after _Andrew Howroyd_ *)
%Y A002831 Cf. A002830 (for not-necessarily connected graphs), A006712, A006713.
%K A002831 nonn
%O A002831 1,2
%A A002831 _N. J. A. Sloane_
%E A002831 a(5) and a(6) corrected and new terms a(7) and a(8) computed by _Sean A. Irvine_, Sep 09 2014
%E A002831 a(9)-a(10) from _Sasha Kolpakov_, Dec 11 2017
%E A002831 a(11) and beyond from _Andrew Howroyd_, Dec 14 2017