A006712 Number of 3-edge-colored trivalent graphs with 2n labeled nodes.

6, 480, 197820, 150474240, 208857587400, 471804812519040, 1625459273858019600, 8112729590064978278400, 56342429224416522460072800, 527075322501595757416502976000, 6466573585901882433727764077860800, 101749747195531624711768653503416320000

Offset: 2

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

Links

• Andrew Howroyd, Table of n, a(n) for n = 2..50
• 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. A006713 (for connected cases), A248361 (for the incorrect values). See also A002830, A002831, A005638.

Programs

PARI

dpermcount(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*=2*t*k;s+=2*t); s!/m}
S(n,x)={vector(n, n, if(n>1, sum(k=0, n, binomial(2*n-k,k)*2*n/(2*n-k)*x^k), 0))}
q(n,s)={my(t=0); if(n>1, forpart(p=n, t+=dpermcount(p)*prod(i=1, #p, s[p[i]]), [2,n])); t}
a(n)={my(p=q(n,S(n,x))); sum(i=0, poldegree(p), polcoeff(p,n-i)*(-1)^(n-i)*(2*i)!/(2^i*i!))} \\ Andrew Howroyd, Dec 18 2017

Extensions

a(5)-a(6) corrected and a(7)-a(10) from Sean A. Irvine, Oct 05 2014

Terms a(11) and beyond from Andrew Howroyd, Dec 18 2017