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

0,3

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]

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).

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 *)

Prog

(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

Crossrefs

Row n=2 of A331126.

Cf. A020554, A020555, A014501, A060053.

Sequence in context: A167016 A300014 A108522 * A101482 A099717 A322772

Adjacent sequences: A014497 A014498 A014499 * A014501 A014502 A014503

Keyword

nonn

Author

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

Status

approved