cp's OEIS Frontend

A001831 Number of labeled graded partially ordered sets with n elements of height at most 1.

%I A001831 M2956 N1194 #60 Mar 08 2021 12:01:17
%S A001831 1,1,3,13,87,841,11643,227893,6285807,243593041,13262556723,
%T A001831 1014466283293,109128015915207,16521353903210521,3524056001906654763,
%U A001831 1059868947134489801413,449831067019305308555487,269568708630308018001547681,228228540531327778410439620963
%N A001831 Number of labeled graded partially ordered sets with n elements of height at most 1.
%C A001831 Labeled posets where for all a,b,c in the set, do not have a<b<c. (Equivalently, labeled posets with no chain of length 3; 3-avoiding posets.)  Labeled digraphs where every node has indegree 0 or outdegree 0.
%C A001831 Number of labeled digraphs with n vertices with no directed path of length 2. Number of n X n {0,1} matrices A such that A^2 = 0. - _Michael Somos_, Jul 28 2013
%C A001831 Number of relations on n labeled nodes that are simultaneously transitive and antitransitive. - _Peter Kagey_, Feb 14 2021
%D A001831 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001831 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001831 T. D. Noe, <a href="/A001831/b001831.txt">Table of n, a(n) for n = 0..50</a>
%H A001831 D. A. Klarner, <a href="/A000798/a000798_11.pdf">The number of graded partially ordered sets</a>, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
%H A001831 D. A. Klarner, <a href="http://dx.doi.org/10.1016/S0021-9800(69)80100-6">The number of graded partially ordered sets</a>, J. Combin. Theory, 6 (1969), 12-19.
%H A001831 A. Motzek and R. Möller, <a href="https://www.ifis.uni-luebeck.de/uploads/tx_wapublications/ai-dbn-motzek-public-ga.pdf">Exploiting Innocuousness in Bayesian Networks</a>, Preprint, Australasian Joint Conference on Artificial Intelligence AI 2015: Advances in Artificial Intelligence, pp. 411-423.
%H A001831 Zvi Rosen and Yan X. Zhang, <a href="https://arxiv.org/abs/1702.06907">Convex Neural Codes in Dimension 1</a>, arXiv:1702.06907 [math.CO], 2017. Mentions this sequence.
%H A001831 <a href="/index/Pos#posets">Index entries for sequences related to posets</a>
%F A001831 a(n) = 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_, Apr 04 2003
%F A001831 E.g.f.: Sum_{n>=0} exp((2^n-1)*x) * x^n/n!. - _Paul D. Hanna_, Nov 27 2007 [correction made by _Paul D. Hanna_, Mar 08 2021]
%F A001831 O.g.f.: Sum_{n>=0} x^n/(1 - (2^n - 1)*x)^(n+1) = Sum_{n>=0} a(n)*x^n. - _Paul D. Hanna_, Sep 15 2009
%F A001831 a(n) ~ c * 2^(n^2/4 + n + 1/2) / sqrt(Pi*n), where c = JacobiTheta3(0,1/2) = EllipticTheta[3, 0, 1/2] = 2.1289368272118771586694585485449... if n is even, and c = JacobiTheta2(0,1/2) = EllipticTheta[2, 0, 1/2] = 2.1289312505130275585916134025753... if n is odd. - _Vaclav Kotesovec_, Mar 10 2014
%e A001831 1 + x + 3*x^2 + 13*x^3 + 87*x^4 + 841*x^5 + 11643*x^6 + 227893*x^7 + ...
%p A001831 A001831 := proc(n)
%p A001831     add(binomial(n,k)*(2^k-1)^(n-k),k=0..n) ;
%p A001831 end proc:
%p A001831 seq(A001831(n),n=0..10) ; # _R. J. Mathar_, Mar 08 2021
%t A001831 Join[{1}, Table[Sum[Binomial[n,k](2^k-1)^(n-k),{k,n}],{n,20}]] (* _Harvey P. Dale_, Jan 05 2012 *)
%o A001831 (PARI) {a(n)=n!*polcoeff(sum(k=0,n,exp((2^k-1)*x)*x^k/k!),n)} \\ _Paul D. Hanna_, 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)} \\ _Paul D. Hanna_, Sep 15 2009
%Y A001831 Cf. A002031, A047863, A052332, A001833, A048194.
%Y A001831 Row sums of A052296.
%Y A001831 Cf. variants: A135753, A135754.
%K A001831 nonn,nice
%O A001831 0,3
%A A001831 _N. J. A. Sloane_
%E A001831 More terms, formula and comments from _Christian G. Bower_, Dec 15 1999
%E A001831 Last 4 terms corrected by _Vladeta Jovovic_, Apr 04 2003
%E A001831 Comments corrected by _Joel B. Lewis_, Mar 28 2011