A014500 Number of graphs with unlabeled (non-isolated) nodes and n labeled edges.

1, 1, 2, 9, 70, 794, 12055, 233238, 5556725, 158931613, 5350854707, 208746406117, 9315261027289, 470405726166241, 26636882237942128, 1678097862705130667, 116818375064650241036, 8932347052564257212796, 746244486452472386213939, 67796741482683128375533560

Offset: 0

Author

Simon Plouffe, Gilbert Labelle (gilbert(AT)lacim.uqam.ca)

Keywords

• nonn

References

• G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100
• P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• Peter Cameron, Thomas Prellberg, Dudley Stark, Asymptotic enumeration of 2-covers and line graphs, Discrete Math. 310 (2010), no. 2, 230-240 (see u_n).
• G. Labelle, Counting enriched multigraphs according to the number of their edges (or arcs), Discrete Math., 217 (2000), 237-248.
• G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]

Crossrefs

Row n=2 of A331126.

Cf. A020554, A020555, A014501, A060053.

Programs

Maple

read("transforms") ;
A020556 := proc(n) local k; add((-1)^(n+k)*binomial(n, k)*combinat[bell](n+k), k=0..n) end proc:
A014500 := proc(n) local i,gexp,lexp;
gexp := [seq(1/2^i/i!,i=0..n+1)] ;
lexp := add( A020556(i)*((log(1+x))/2)^i/i!,i=0..n+1) ;
lexp := taylor(lexp,x=0,n+1) ;
lexp := gfun[seriestolist](lexp,'ogf') ;
CONV(gexp,lexp) ; op(n+1,%)*n! ; end proc:
seq(A014500(n),n=0..20) ; # R. J. Mathar, Jul 03 2011

Mathematica

max = 20; A020556[n_] := Sum[(-1)^(n+k)*Binomial[n, k]*BellB[n+k], {k, 0, n}]; egf = Exp[x/2]*Sum[A020556[n]*(Log[1+x]/2)^n/n!, {n, 0, max}] + O[x]^max; CoefficientList[egf, x]*Range[0, max-1]! (* Jean-François Alcover, Feb 19 2017, after Vladeta Jovovic *)

PARI

\\ here egf1 is A020556 as e.g.f.
egf1(n)={my(bell=serlaplace(exp(exp(x + O(x^(2*n+1)))-1))); sum(i=0, n, sum(k=0, i, (-1)^k*binomial(i,k)*polcoef(bell, 2*i-k))*x^i/i!) + O(x*x^n)}
seq(n)={my(B=egf1(n), L=log(1+x + O(x*x^n))/2); Vec(serlaplace(exp(x/2 + O(x*x^n))*sum(k=0, n, polcoef(B ,k)*L^k)))} \\ Andrew Howroyd, Jan 13 2020

Formula

E.g.f.: exp(-1+x/2)*Sum((1+x)^binomial(n, 2)/n!, n=0..infinity) [probably in the Labelle paper]. - Vladeta Jovovic, Apr 27 2004

E.g.f.: exp(x/2)*Sum(A020556(n)*(log(1+x)/2)^n/n!, n=0..infinity). - Vladeta Jovovic, May 02 2004

Binomial transform of A060053.

The e.g.f.'s of A020554 (S(x)) and A014500 (U(x)) are related by S(x) = U(e^x-1).

The e.g.f.'s of A014500 (U(x)) and A060053 (V(x)) are related by U(x) = e^x*V(x).