Original entry on oeis.org
1, 1, 7, 73, 1051, 19381, 436087, 11585953, 354981571, 12322179901, 477938035807, 20485584143113, 961567521142411, 49054912287659461, 2702571588828034567, 159911968233095867953, 10114120854154243738771, 680943323845807848142861
Offset: 1
A048172
Number of labeled series-parallel graphs with n edges.
Original entry on oeis.org
1, 3, 19, 195, 2791, 51303, 1152019, 30564075, 935494831, 32447734143, 1257770533339, 53884306900515, 2528224238464471, 128934398091500823, 7101273378743303779, 420078397130637237915, 26563302733186339752511, 1788055775343964413724143, 127652707703771090396080939
Offset: 1
- 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.
- R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.39.
- Vincenzo Librandi, Table of n, a(n) for n = 1..100
- F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
- Frédéric Fauvet, L. Foissy, D. Manchon, Operads of finite posets, arXiv preprint arXiv:1604.08149 [math.CO], 2016.
- S. R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
- Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.
- R. P. Stanley, Enumeration of posets generated by disjoint unions and ordinal sums, Proc. Amer. Math. Soc. 45 (1974), 295-299
- Index entries for reversions of series
- Index entries for sequences related to posets
-
with(gfun):
f := series((ln(1+x)-x^2/(1+x)), x, 21):
egf := seriestoseries(f, 'revogf'):
seriestolist(egf, 'Laplace');
-
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. *)
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 *)
-
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 */
-
x='x+O('x^55);
s=-log(1-x)-x^2/(1-x);
A048174=Vec(serlaplace(serreverse(s)));
t=x+2*(1-cosh(x));
A058349=Vec(serlaplace(serreverse(t)));
A048172=A048174+A058349; A048172[1]-=1;
A048172 /* Joerg Arndt, Feb 04 2011 */
A058349
Number of connected labeled series-parallel posets on n nodes.
Original entry on oeis.org
1, 2, 12, 122, 1740, 31922, 715932, 18978122, 580513260, 20125554242, 779832497532, 33398722757402, 1566656717322060, 79879485803841362, 4398701789915269212, 260166428897541369962, 16449181879032096013740, 1107112451498156565581282, 79030557433744270179981372
Offset: 1
- R. C. Read, Graphical enumeration by cycle-index sums: first steps toward a unified treatment, preprint, Sept. 26, 1991.
- R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.39, page 133, g(n).
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(continue from A053554) t1 := log(1+EGF053554): t2 := series(t1,x,30); SERIESTOLISTMULT(t2);
-
Drop[ CoefficientList[ InverseSeries[ Series[x + 2*(1 - Cosh[x]) , {x, 0, 19}], y], y], 1]* Range[19]! (* Jean-François Alcover, Sep 21 2011, after g.f. *)
-
a(n):=if n=1 then 1 else (n-1)!*sum(binomial(n+k-1,n-1)*sum(binomial(k,j)*((sum((binomial(j,l)*((-1)^(n-l+j-1)+1)*sum(binomial(j-l,r)*2^(j-l-r-1)*(-1)^(r-j)*sum((r-2*i)^(n-l+j-1)*binomial(r,i),i,0,r),r,1,j-l))/(n-l+j-1)!,l,0,j-1))),j,1,k),k,1,n-1); /* Vladimir Kruchinin, Feb 19 2012 */
-
/* Joerg Arndt, Feb 04 2011 */
x='x+O('x^55); t=x+2*(1-cosh(x));
Vec(serlaplace(serreverse(t))) /* show terms */
A339300
Number of essentially parallel oriented series-parallel networks with n labeled elements and without multiple unit elements in parallel.
Original entry on oeis.org
1, 0, 6, 36, 540, 8400, 169680, 3966480, 107518320, 3295283040, 112888369440, 4272403544640, 177061349424960, 7974538914101760, 387840385867334400, 20257533315635616000, 1130954856127948051200, 67208532822729871372800, 4235759061057115720128000
Offset: 1
A048174 is the case with multiple edges in parallel allowed.
A058379 is the case that order is not significant in series configurations.
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seq(n, Z=x)={my(p=Z+O(x^2)); for(n=2, n, p = (1 + Z)*exp(p^2/(1+p)) - 1); Vec(serlaplace(1-1/(1+p)))}
Showing 1-4 of 4 results.
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