A006712 - OEIS

%I M4311 #25 Dec 18 2017 22:44:55

%S 6,480,197820,150474240,208857587400,471804812519040,

%T 1625459273858019600,8112729590064978278400,

%U 56342429224416522460072800,527075322501595757416502976000,6466573585901882433727764077860800,101749747195531624711768653503416320000

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

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

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

%H Andrew Howroyd, <a href="/A006712/b006712.txt">Table of n, a(n) for n = 2..50</a>

%H Sean A. Irvine, <a href="/A006712/a006712.pdf">Illustration of initial terms</a>

%H R. C. Read, <a href="/A002831/a002831.pdf">Letter to N. J. A. Sloane, Feb 04 1971</a> (gives initial terms of this sequence)

%o (PARI)

%o 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}

%o 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))}

%o 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}

%o 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

%Y Cf. A006713 (for connected cases), A248361 (for the incorrect values). See also A002830, A002831, A005638.

%K nonn

%O 2,1

%A _N. J. A. Sloane_

%E a(5)-a(6) corrected and a(7)-a(10) from _Sean A. Irvine_, Oct 05 2014

%E Terms a(11) and beyond from _Andrew Howroyd_, Dec 18 2017