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

%I A014500 #51 Jan 13 2020 13:53:46
%S A014500 1,1,2,9,70,794,12055,233238,5556725,158931613,5350854707,
%T A014500 208746406117,9315261027289,470405726166241,26636882237942128,
%U A014500 1678097862705130667,116818375064650241036,8932347052564257212796,746244486452472386213939,67796741482683128375533560
%N A014500 Number of graphs with unlabeled (non-isolated) nodes and n labeled edges.
%D A014500 G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
%H A014500 Alois P. Heinz, <a href="/A014500/b014500.txt">Table of n, a(n) for n = 0..100</a>
%H A014500 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A014500 Peter Cameron, Thomas Prellberg, Dudley Stark, <a href="http://dx.doi.org/10.1016/j.disc.2008.09.008">Asymptotic enumeration of 2-covers and line graphs</a>, Discrete Math. 310 (2010), no. 2, 230-240 (see u_n).
%H A014500 G. Labelle, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00265-4">Counting enriched multigraphs according to the number of their edges (or arcs)</a>, Discrete Math., 217 (2000), 237-248.
%H A014500 G. Paquin, <a href="/A038205/a038205.pdf">Dénombrement de multigraphes enrichis</a>, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]
%F A014500 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
%F A014500 E.g.f.: exp(x/2)*Sum(A020556(n)*(log(1+x)/2)^n/n!, n=0..infinity). - _Vladeta Jovovic_, May 02 2004
%F A014500 Binomial transform of A060053.
%F A014500 The e.g.f.'s of A020554 (S(x)) and A014500 (U(x)) are related by S(x) = U(e^x-1).
%F A014500 The e.g.f.'s of A014500 (U(x)) and A060053 (V(x)) are related by U(x) = e^x*V(x).
%p A014500 read("transforms") ;
%p A014500 A020556 := proc(n) local k; add((-1)^(n+k)*binomial(n, k)*combinat[bell](n+k), k=0..n) end proc:
%p A014500 A014500 := proc(n) local i,gexp,lexp;
%p A014500 gexp := [seq(1/2^i/i!,i=0..n+1)] ;
%p A014500 lexp := add( A020556(i)*((log(1+x))/2)^i/i!,i=0..n+1) ;
%p A014500 lexp := taylor(lexp,x=0,n+1) ;
%p A014500 lexp := gfun[seriestolist](lexp,'ogf') ;
%p A014500 CONV(gexp,lexp) ; op(n+1,%)*n! ; end proc:
%p A014500 seq(A014500(n),n=0..20) ; # _R. J. Mathar_, Jul 03 2011
%t A014500 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_ *)
%o A014500 (PARI) \\ here egf1 is A020556 as e.g.f.
%o A014500 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)}
%o A014500 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
%Y A014500 Row n=2 of A331126.
%Y A014500 Cf. A020554, A020555, A014501, A060053.
%K A014500 nonn
%O A014500 0,3
%A A014500 _Simon Plouffe_, Gilbert Labelle (gilbert(AT)lacim.uqam.ca)