A002831 - OEIS

%I M3424 N1388 #82 Aug 26 2022 10:35:40

%S 1,4,11,60,318,2806,29359,396196,6231794,112137138,2249479114,

%T 49691965745,1197158348160,31230408793660,876971159096883,

%U 26374570956403684,845812191249484022,28812214090645864661,1038982259432805270094,39540452134474760212909

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

%C 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 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 From _Sasha Kolpakov_, Dec 17 2017: (Start)

%C Number of oriented unrooted pavings (after Arques & Koch, Spehner, Lienhardt) with 2n darts.

%C Also the number of conjugacy classes of free index 2n subgroups in the free product Z_2*Z_2*Z_2. (End)

%D R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Andrew Howroyd, <a href="/A002831/b002831.txt">Table of n, a(n) for n = 1..30</a>

%H 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 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 R. C. Read, <a href="/A002831/a002831.pdf">Letter to N. J. A. Sloane, Feb 04 1971</a>

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerTransform.html">Euler Transform</a>

%F 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 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 terms = 20;

%t 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 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 pm[v_] := Module[{p = Total[x^v]}, Product[ b[Coefficient[p, x, i], i], {i, 1, Exponent[p, x]}]];

%t a2830[n_] := Module[{s = 0}, Do[ s += permcount[p] pm[p]^3, {p, IntegerPartitions[2 n]}]; s/(2 n)!];

%t G[x_] = 1 + Sum[a2830[n] x^n, {n, 1, terms+1}];

%t gf = Sum[MoebiusMu[k] Log[G[x^k]]/k, {k, 1, terms+1}] + O[x]^(terms+1);

%t CoefficientList[gf, x] // Rest (* _Jean-François Alcover_, Jul 02 2018, after _Andrew Howroyd_ *)

%Y Cf. A002830 (for not-necessarily connected graphs), A006712, A006713.

%K nonn

%O 1,2

%A _N. J. A. Sloane_

%E a(5) and a(6) corrected and new terms a(7) and a(8) computed by _Sean A. Irvine_, Sep 09 2014

%E a(9)-a(10) from _Sasha Kolpakov_, Dec 11 2017

%E a(11) and beyond from _Andrew Howroyd_, Dec 14 2017