A002830 Number of 3-edge-colored trivalent graphs with 2n nodes.

1, 1, 5, 16, 86, 448, 3580, 34981, 448628, 6854130, 121173330, 2403140605, 52655943500, 1260724587515, 32726520985365, 915263580719998, 27432853858637678, 877211481667946811, 29807483816421710806, 1072542780403547030073, 40739888428757581326987

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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 = 0..200
• Nicholas Dub, Enumeration of triangulations modulo symmetries and of rooted triangulations counted by their number of (d - 2)-simplices in dimension d ≥ 2, tel-03641958 [cs.OH], Université Paris-Nord - Paris XIII, 2021.
• Sean A. Irvine, Illustration of initial terms
• R. C. Read, Letter to N. J. A. Sloane, Feb 04 1971 (gives initial terms of this sequence)

Crossrefs

Cf. A002831, A006712, A006713.

Programs

Mathematica

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]}]];
a[n_] := Module[{s = 0}, Do[s += permcount[p] pm[p]^3, {p, IntegerPartitions[2 n]}]; s/(2 n)!];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 30}] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)

PARI

b(k,r) = {if(k%2, if(r%2, 0, my(j=r/2); k^j*(2*j)!/(j!*2^j)), sum(j=0, r\2, binomial(r, 2*j)*k^j*(2*j)!/(j!*2^j)))}
g(n,k)={sum(r=0, n\k,  x^(k*r)*b(k,r)^3/(r!*k^r)) + O(x*x^n)}
seq(n)={Vec(substpol(prod(k=1, 2*n, g(2*n,k)), x^2, x))} \\ Andrew Howroyd, Dec 14 2017; updated May 02 2023

Formula

G.f.: exp(Sum_{k >= 1} F(x^k) / k) where F(x) is the g.f. for A002831. - Sean A. Irvine, Sep 09 2014

Extensions

a(7)-a(8) from Sean A. Irvine, Sep 08 2014

Terms a(9) and beyond from Andrew Howroyd, Dec 14 2017

a(0)=1 prepended by Andrew Howroyd, May 02 2023