A002830 Number of 3-edge-colored trivalent graphs with 2n nodes. (Formerly M3871 N1586)

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

Offset

0,3

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)

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

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 *)

Prog

(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

Crossrefs

Cf. A002831, A006712, A006713.

Sequence in context: A179685 A286077 A286072 * A365654 A370096 A196015

Adjacent sequences: A002827 A002828 A002829 * A002831 A002832 A002833

Keyword

nonn

Author

N. J. A. Sloane

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

Status

approved