%I #71 Jan 14 2018 22:59:06

%S 1,3,19,195,2791,51303,1152019,30564075,935494831,32447734143,

%T 1257770533339,53884306900515,2528224238464471,128934398091500823,

%U 7101273378743303779,420078397130637237915,26563302733186339752511,1788055775343964413724143,127652707703771090396080939

%N Number of labeled series-parallel graphs with n edges.

%C Labeled N-free posets. - Detlef Pauly (dettodet(AT)yahoo.de), Dec 27 2002

%D Ronald C. Read, Graphical enumeration by cycle-index sums: first steps toward a unified treatment, Research Report CORR 91-19, University of Waterloo, Sept 1991.

%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.39.

%H Vincenzo Librandi, <a href="/A048172/b048172.txt">Table of n, a(n) for n = 1..100</a>

%H F. Chapoton, F. Hivert, J.-C. Novelli, <a href="http://arxiv.org/abs/1307.0092">A set-operad of formal fractions and dendriform-like sub-operads</a>, arXiv preprint arXiv:1307.0092 [math.CO], 2013.

%H Frédéric Fauvet, L. Foissy, D. Manchon, <a href="https://arxiv.org/abs/1604.08149">Operads of finite posets</a>, arXiv preprint arXiv:1604.08149 [math.CO], 2016.

%H S. R. Finch, <a href="/A000084/a000084_2.pdf">Series-parallel networks</a>, July 7, 2003. [Cached copy, with permission of the author]

%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1211.3244">The method for obtaining expressions for coefficients of reverse generating functions</a>, arXiv:1211.3244 [math.CO], 2012.

%H R. P. Stanley, <a href="http://www.ams.org/journals/proc/1974-045-02/S0002-9939-1974-0351928-7/">Enumeration of posets generated by disjoint unions and ordinal sums</a>, Proc. Amer. Math. Soc. 45 (1974), 295-299

%H <a href="/index/Res#revert">Index entries for reversions of series</a>

%H <a href="/index/Pos#posets">Index entries for sequences related to posets</a>

%F a(n) = A058349(n) + A048174(n).

%F a(n) = A058349(n) + A058350(n) (n>=2).

%F Reference (by Ronald C. Read) gives generating functions.

%F E.g.f. is reversion of log(1+x)-x^2/(1+x).

%F a(n)=if n=1 then 1 else -sum((k!/n!*stirling1(n,k)+sum(binomial(k,j)*sum((j)!/(i)!*stirling1(i,j)*h(n-i,k-j),i,j,n-k+j),j,1,k-1)+h(n,k))*a(k),k,1,n-1), h(n,k)=if n=k then 0 else (-1)^(n-k)*binomial(n-k-1,k-1), n>0. - _Vladimir Kruchinin_, Sep 08 2010

%F a(n) ~ sqrt((5+3*sqrt(5))/10) * n^(n-1) / (exp(n) * (2 - sqrt(5) + log((1+sqrt(5))/2))^(n-1/2)). - _Vaclav Kotesovec_, Feb 25 2014

%p with(gfun):

%p f := series((ln(1+x)-x^2/(1+x)), x, 21):

%p egf := seriestoseries(f, 'revogf'):

%p seriestolist(egf, 'Laplace');

%t lim = 19; Join[{1}, Drop[ CoefficientList[ InverseSeries[ Series[x + 2*(1 - Cosh[x]) , {x, 0, lim}], y] + InverseSeries[ Series[-Log[1 - x] - x^2/(1 - x),{x, 0, lim}], y], y], 2]*Range[2, lim]!] (* _Jean-François Alcover_, Sep 21 2011, after g.f. *)

%t m = 17; Rest[CoefficientList[InverseSeries[Series[Log[1+x]-x^2/(1+x), {x, 0, m}], x], x]*Table[k!,{k, 0, m}]](* _Jean-François Alcover_, Apr 18 2011 *)

%o (PARI)

%o x='x+O('x^55);

%o s=-log(1-x)-x^2/(1-x);

%o A048174=Vec(serlaplace(serreverse(s)));

%o t=x+2*(1-cosh(x));

%o A058349=Vec(serlaplace(serreverse(t)));

%o A048172=A048174+A058349; A048172[1]-=1;

%o A048172 /* _Joerg Arndt_, Feb 04 2011 */

%o (Maxima) h(n,k):=if n=k then 0 else (-1)^(n-k)*binomial(n-k-1,k-1); a(n):=if n=1 then 1 else -sum((k!/n!*stirling1(n,k)+sum(binomial(k,j)*sum((j)!/(i)!*stirling1(i,j)*h(n-i,k-j),i,j,n-k+j),j,1,k-1)+h(n,k))*a(k),k,1,n-1); /* _Vladimir Kruchinin_, Sep 08 2010 */

%Y Cf. A048174, A058349, A058350.

%Y Cf. A000112 (unlabeled posets), A001035 (labeled posets), A003430 (unlabeled analog).

%K nonn,easy,nice

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Joerg Arndt_, Feb 04 2011