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A002830 Number of 3-edge-colored trivalent graphs with 2n nodes.
(Formerly M3871 N1586)
4
1, 5, 16, 86, 448, 3580, 34981, 448628, 6854130, 121173330, 2403140605, 52655943500, 1260724587515, 32726520985365, 915263580719998, 27432853858637678, 877211481667946811, 29807483816421710806, 1072542780403547030073, 40739888428757581326987 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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(F(x^k) / k, k >= 1) 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-Fran├žois Alcover, Jul 02 2018, after Andrew Howroyd *)

PROG

(PARI)

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, 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(q%2, if(k%2, 0, my(j=k/2); q^j*(2*j)!/(j!*2^j)), sum(j=0, k\2, binomial(k, 2*j)*q^j*(2*j)!/(j!*2^j)))}

pm(v) = {my(p=sum(i=1, #v, x^v[i])); prod(i=1, poldegree(p), b(polcoeff(p, i), i))}

a(n) = {my(s=0); forpart(p=2*n, s+=permcount(p)*pm(p)^3); s/(2*n)!} \\ Andrew Howroyd, Dec 14 2017

CROSSREFS

Cf. A002831, A006712, A006713.

Sequence in context: A179685 A286077 A286072 * A196015 A332625 A307953

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

STATUS

approved

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Last modified November 26 16:27 EST 2020. Contains 338641 sequences. (Running on oeis4.)