A002831 Number of 3-edge-colored connected trivalent graphs with 2n nodes. (Formerly M3424 N1388)

1, 4, 11, 60, 318, 2806, 29359, 396196, 6231794, 112137138, 2249479114, 49691965745, 1197158348160, 31230408793660, 876971159096883, 26374570956403684, 845812191249484022, 28812214090645864661, 1038982259432805270094, 39540452134474760212909

Offset

1,2

Comments

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

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.

From Sasha Kolpakov, Dec 17 2017: (Start)

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

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

References

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

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

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

Links

Andrew Howroyd, Table of n, a(n) for n = 1..30

Rémi Bottinelli, Laura Ciobanu, and Alexander Kolpakov, Three-dimensional maps and subgroup growth, manuscripta math. (2021).

L. Ciobanu and A. Kolpakov, Three-dimensional maps and subgroup growth, arXiv:1712.01418 [math.GR], 2017.

R. C. Read, Letter to N. J. A. Sloane, Feb 04 1971

Neriman Tokcan, Jonathan Gryak, Kayvan Najarian, and Harm Derksen, Algebraic Methods for Tensor Data, arXiv:2005.12988 [math.RT], 2020.

Eric Weisstein's World of Mathematics, Euler Transform

Formula

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

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

Mathematica

terms = 20;
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];
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]}]];
pm[v_] := Module[{p = Total[x^v]}, Product[ b[Coefficient[p, x, i], i], {i, 1, Exponent[p, x]}]];
a2830[n_] := Module[{s = 0}, Do[ s += permcount[p] pm[p]^3, {p, IntegerPartitions[2 n]}]; s/(2 n)!];
G[x_] = 1 + Sum[a2830[n] x^n, {n, 1, terms+1}];
gf = Sum[MoebiusMu[k] Log[G[x^k]]/k, {k, 1, terms+1}] + O[x]^(terms+1);
CoefficientList[gf, x] // Rest (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)

Crossrefs

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

Sequence in context: A203577 A081073 A245545 * A246598 A242749 A303955

Adjacent sequences: A002828 A002829 A002830 * A002832 A002833 A002834

Keyword

nonn

Author

N. J. A. Sloane

Extensions

a(5) and a(6) corrected and new terms a(7) and a(8) computed by Sean A. Irvine, Sep 09 2014

a(9)-a(10) from Sasha Kolpakov, Dec 11 2017

a(11) and beyond from Andrew Howroyd, Dec 14 2017

Status

approved