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A206950 Number of nonisomorphic graded posets with 0 and non-uniform Hasse graph of rank n, with no 3-element antichain.

%I A206950 #33 May 17 2021 08:49:26
%S A206950 0,0,0,3,33,259,1762,11093,66592,387264,2202053,12314587,67995221,
%T A206950 371697914,2015659707,10859379024,58190011080,310409500291,
%U A206950 1649579166385,8738000970251,46158910515154,243260704208613,1279386591175904,6716811592446952,35209193397256085
%N A206950 Number of nonisomorphic graded posets with 0 and non-uniform Hasse graph of rank n, with no 3-element antichain.
%C A206950 We do not assume all maximal elements have maximal rank and thus use graded poset to mean: For every element x, all maximal chains among those with x as greatest element have the same finite length. Here, the term uniform used in the sense of Retakh, Serconek and Wilson.
%H A206950 Vincenzo Librandi, <a href="/A206950/b206950.txt">Table of n, a(n) for n = 0..1000</a>
%H A206950 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 A206950 Wikipedia, <a href="http://en.wikipedia.org/wiki/Graded_poset">Graded poset</a>
%H A206950 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (13,-59,115,-109,51,-9).
%F A206950 a(n) = 13*a(n-1) - 59*a(n-2) + 115*a(n-3) - 109*a(n-4) + 51*a(n-5) - 9*a(n-6), a(1)=0, a(2)=0, a(3)=3, a(4)=33, a(5)=259, a(6)=1762.
%F A206950 G.f.: (3*x^3-6*x^4+7*x^5-3*x^6)/((-1+7*x-10*x^2+3*x^3)*(-1+6*x-7*x^2+3*x^3)).
%t A206950 Join[{0},LinearRecurrence[{13, -59, 115, -109, 51, -9}, {0, 0, 3, 33, 259, 1762}, 40]]
%o A206950 (Python)
%o A206950 def a(n,adict={0:0,1:0,2:0,3:3,4:33,5:259,6:1762}):
%o A206950     if n in adict:
%o A206950         return adict[n]
%o A206950     adict[n]=13*a(n-1)-59*a(n-2)+115*a(n-3)-109*a(n-4)+51*a(n-5)-9*a(n-6)
%o A206950     return adict[n]
%Y A206950 A206901(n)-A206902(n).
%Y A206950 Cf. A206949 (unique maximal element added.)
%Y A206950 Cf. A206947, A206948 (requiring exactly two elements in each rank level above 0 with and without maximal element.)
%K A206950 nonn,easy
%O A206950 0,4
%A A206950 _David Nacin_, Feb 13 2012