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A206949 Number of nonisomorphic graded posets with 0 and non-uniform Hasse graph of rank n, with no 3-element antichain.

%I A206949 #36 May 17 2021 08:49:19
%S A206949 0,0,0,3,24,135,657,2961,12744,53244,218025,880308,3518721,13961727,
%T A206949 55097091,216546048,848476296,3316800555,12942852624,50437433079,
%U A206949 196347606849,763752142233,2969021213928,11536374392820,44809232564673,173997851613660,675501426136017
%N A206949 Number of nonisomorphic graded posets with 0 and non-uniform Hasse graph of rank n, with no 3-element antichain.
%C A206949 Here, the term uniform is used in the sense of Retakh, Serconek and Wilson.  Graded is used in terms of Stanley's definition that all maximal chains have the same length n.
%D A206949 Richard P. Stanley, Enumerative combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.
%H A206949 Vincenzo Librandi, <a href="/A206949/b206949.txt">Table of n, a(n) for n = 0..1000</a>
%H A206949 V. Retakh, S. Serconek and R. Wilson, <a href="http://arxiv.org/abs/1010.6295">Hilbert Series of Algebras Associated to Directed Graphs and Order Homology</a>, arXiv:1010.6295 [math.RA], 2010-2011.
%H A206949 Wikipedia, <a href="http://en.wikipedia.org/wiki/Graded_poset">Graded poset</a>
%H A206949 <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (9,-27,30,-9).
%F A206949 a(n) = 9*a(n-1) - 27*a(n-2) + 30*a(n-3) - 9*a(n-4), a(1)=0, a(2)=0, a(3)=3, a(4)=24.
%F A206949 G.f.: (3*(1-x)*x^3)/((1-3*x)*(1-6*x+9*x^2-3*x^3)).
%F A206949 a(n) = A124292(n+1) - A025192(n).
%t A206949 Join[{0}, LinearRecurrence[{9, -27, 30, -9}, {0, 0, 3, 24}, 40]]
%o A206949 (Python)
%o A206949 def a(n,adict={0:0,1:0,2:0,3:3,4:24}):
%o A206949     if n in adict:
%o A206949         return adict[n]
%o A206949     adict[n]=9*a(n-1)-27*a(n-2)+30*a(n-3)-9*a(n-4)
%o A206949     return adict[n]
%Y A206949 Cf. A206950 (maximal element removed).
%Y A206949 Cf. A206947, A206948 (requiring exactly two elements in each rank level above 0 with and without maximal element).
%K A206949 nonn,easy
%O A206949 0,4
%A A206949 _David Nacin_, Feb 13 2012