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 A048174 Number of labeled chains with n edges. 5
 1, 1, 7, 73, 1051, 19381, 436087, 11585953, 354981571, 12322179901, 477938035807, 20485584143113, 961567521142411, 49054912287659461, 2702571588828034567, 159911968233095867953, 10114120854154243738771, 680943323845807848142861, 48622150270026820216099567, 3670113810844512283440027673 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 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 x1. 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 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 *) PROG (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 CROSSREFS Sequence in context: A250917 A112939 A058350 * A352118 A258379 A325930 Adjacent sequences: A048171 A048172 A048173 * A048175 A048176 A048177 KEYWORD nonn,easy,nice AUTHOR N. J. A. Sloane EXTENSIONS More terms from Joerg Arndt, Feb 04 2011 STATUS approved

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Last modified April 15 18:46 EDT 2024. Contains 371696 sequences. (Running on oeis4.)