A048174 Number of labeled chains with n edges.

1, 1, 7, 73, 1051, 19381, 436087, 11585953, 354981571, 12322179901, 477938035807, 20485584143113, 961567521142411, 49054912287659461, 2702571588828034567, 159911968233095867953, 10114120854154243738771, 680943323845807848142861, 48622150270026820216099567, 3670113810844512283440027673

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Number of labeled series-parallel posets on n nodes that are not a nontrivial ordinal sum.

Let ( T, < ) and ( U, < ) be posets with T and U disjoint. Their ordinal sum is ( T union U, < ) where x

References

• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.39, page 133, h(n).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..100
• 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, Enumeration of posets generated by disjoint unions and ordinal sums, Proc. Amer. Math. Soc. 45 (1974), 295-299
• Index entries for sequences related to posets

Programs

Maple

with(gfun): f := series(ln(1+x)-x^2/(1+x), x, 30):
egf := seriestoseries(f, 'revogf'):
t := series(egf/(1+egf), x, 21):
seriestolist(t, 'Laplace');

Mathematica

lim = 20; Drop[ CoefficientList[ InverseSeries[ Series[-Log[1 - x] - x^2/(1 - x), {x, 0, lim}], y], y], 1]*Range[lim]! (* Jean-François Alcover, Sep 21 2011, after g.f. *)
max = 18; S053554 = InverseSeries[ Series[ Log[1+x] - x^2/(1+x), {x, 0, max}], x]; Drop[ CoefficientList[ Series[ S053554 / (1+S053554), {x, 0, max}], x]* Range[0, max]!, 1] (* Jean-François Alcover, Nov 29 2011, after Maple *)

Maxima

a(n):=if n=1 then 1 else (sum((n+k-1)!*sum((-1)^(j)/(k-j)!*((sum(sum((binomial(-l+i-1,l-1)*(-1)^(n-i-1)*stirling1(n+j-i-1,j-l))/(l!*(n+j-i-1)!),i,2*l,n+l-1),l,1,j))+((-1)^(n-1)*stirling1(n+j-1,j))/(n+j-1)!),j,1,k),k,1,n-1)); /* Vladimir Kruchinin, Feb 19 2012 */

PARI

x='x+O('x^66); s=serreverse(log(1+x)-x^2/(1+x)); Vec(serlaplace(s/(1+s))) \\ Joerg Arndt, Mar 11 2014

Formula

Reference gives generating functions (see PARI code for one example).

A048172(n) = A058349(n) + a(n), n>1.

A053554(n) = A058349(n) + A058350(n) (n>=2).

a(n)=sum(k=1..n-1, (n+k-1)!*sum(j=1..k, (-1)^(j)/(k-j)!*((sum(l=1..j, sum(i=2*l..n+l-1, (binomial(-l+i-1,l-1)*(-1)^(n-i-1)*stirling1(n+j-i-1,j-l))/(l!*(n+j-i-1)!))))+((-1)^(n-1)*stirling1(n+j-1,j))/(n+j-1)!))). - Vladimir Kruchinin, Feb 19 2012

a(n) ~ (5-sqrt(5)) * n^(n-1) / (2*5^(3/4)*exp(n)*(2-sqrt(5)+log((1+sqrt(5))/2))^(n-1/2)). - Vaclav Kotesovec, Mar 09 2014

Extensions

More terms from Joerg Arndt, Feb 04 2011

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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