%I A001831 M2956 N1194
%S A001831 1,1,3,13,87,841,11643,227893,6285807,243593041,13262556723,
%T A001831 1014466283293,109128015915207,16521353903210521,3524056001906654763,
%U A001831 1059868947134489801413,449831067019305308555487
%N A001831 Number of labeled graded partially ordered sets with n elements.
%C A001831 Labeled posets where for all i,j,k in the set, do not have a<b<c. Labeled 
               digraphs where every node has indegree 0 or outdegree 0.
%D A001831 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001831 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001831 D. A. Klarner, The number of graded partially ordered sets, J. Combin. 
               Theory, 6 (1969), 12-19.
%H A001831 <a href="Sindx_Pos.html#posets">Index entries for sequences related to 
               posets</a>
%F A001831 Sum((-1)^k*C(n, k)*A047863(k), k=0..n).
%F A001831 a(n) = Sum_{k=0..n} binomial(n, k)*(2^k-1)^(n-k). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Apr 04 2003
%F A001831 E.g.f. A(x) = Sum_{n>=0} exp((2^k-1)*x)*x^n/n!. - Paul D. Hanna (pauldhanna(AT)juno.com), 
               Nov 27 2007
%F A001831 O.g.f.: Sum_{n>=0} x^n/(1 - (2^n - 1)*x)^(n+1) = Sum_{n>=0} a(n)*x^n. 
               [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 15 2009]
%o A001831 (PARI) {a(n)=n!*polcoeff(sum(k=0,n,exp((2^k-1)*x)*x^k/k!),n)} - Paul 
               D. Hanna (pauldhanna(AT)juno.com), Nov 27 2007
%o A001831 (PARI) {a(n)=polcoeff(sum(k=0, n, x^k/(1-(2^k-1)*x +x*O(x^n))^(k+1)), 
               n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 15 2009]
%Y A001831 Cf. A002031, A047863, A052332, A001833, A048194.
%Y A001831 Row sums of A052296.
%Y A001831 Cf. variants: A135753, A135754.
%Y A001831 Sequence in context: A157451 A152112 A054420 this_sequence A002725 A097711 
               A114477
%Y A001831 Adjacent sequences: A001828 A001829 A001830 this_sequence A001832 A001833 
               A001834
%K A001831 nonn,nice
%O A001831 0,3
%A A001831 N. J. A. Sloane (njas(AT)research.att.com).
%E A001831 More terms, formula and comments from Christian G. Bower (bowerc(AT)usa.net), 
               Dec 15 1999.
%E A001831 Last 4 terms corrected by Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 
               04 2003