A058349 - OEIS

%I #27 Nov 17 2024 19:13:36

%S 1,2,12,122,1740,31922,715932,18978122,580513260,20125554242,

%T 779832497532,33398722757402,1566656717322060,79879485803841362,

%U 4398701789915269212,260166428897541369962,16449181879032096013740,1107112451498156565581282,79030557433744270179981372

%N Number of connected labeled series-parallel posets on n nodes.

%C Also, number of labeled blobs with n edges.

%D R. C. Read, Graphical enumeration by cycle-index sums: first steps toward a unified treatment, preprint, Sept. 26, 1991.

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

%H Vincenzo Librandi, <a href="/A058349/b058349.txt">Table of n, a(n) for n = 1..100</a>

%H R. P. Stanley, <a href="https://dx.doi.org/10.1090/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/Pos#posets">Index entries for sequences related to posets</a>

%F Read (1991) reference gives generating functions (see PARI code for one example).

%F A048172(n) = a(n)+A048174(n), n>1.

%F a(n) = (n-1)!*sum(k=1..n-1, binomial(n+k-1,n-1)*sum(j=1..k, binomial(k,j)*((sum(l=0..j-1, (binomial(j,l)*((-1)^(n-l+j-1)+1)*sum(r=1..j-l, binomial(j-l,r)*2^(j-l-r-1)*(-1)^(r-j)*sum(i=0..r, (r-2*i)^(n-l+j-1)*binomial(r,i))))/(n-l+j-1)!))))), n>1, a(1)=1. - _Vladimir Kruchinin_, Feb 19 2012

%F a(n) ~ n^(n-1) / (5^(1/4)*exp(n)*(2-sqrt(5)+log((1+sqrt(5))/2))^(n-1/2)). - _Vaclav Kotesovec_, Mar 09 2014

%p (continue from A053554) t1 := log(1+EGF053554): t2 := series(t1,x,30); SERIESTOLISTMULT(t2);

%t 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. *)

%o (PARI) /* Joerg Arndt, Feb 04 2011 */

%o x='x+O('x^55); t=x+2*(1-cosh(x));

%o Vec(serlaplace(serreverse(t))) /* show terms */

%o (Maxima) 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 */

%Y A053554(n) = a(n) + A058350(n) (n>=2).

%K nonn,easy,nice

%O 1,2

%A _N. J. A. Sloane_, Dec 16 2000

%E More terms from _Joerg Arndt_, Feb 04 2011