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A014500
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Number of graphs with unlabeled (non-isolated) nodes and n labeled edges.
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20
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1, 1, 2, 9, 70, 794, 12055, 233238, 5556725, 158931613, 5350854707, 208746406117, 9315261027289, 470405726166241, 26636882237942128, 1678097862705130667, 116818375064650241036, 8932347052564257212796, 746244486452472386213939, 67796741482683128375533560
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OFFSET
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0,3
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REFERENCES
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G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
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LINKS
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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]
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FORMULA
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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).
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Simon Plouffe, Gilbert Labelle (gilbert(AT)lacim.uqam.ca)
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STATUS
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approved
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