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A208736 Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank level between 0 and 1.

%I A208736 #31 May 21 2021 08:09:41
%S A208736 0,0,0,1,5,22,91,361,1392,5265,19653,72694,267179,977593,3565600,
%T A208736 12975457,47142021,171075606,620303547,2247803785,8141857808,
%U A208736 29481675889,106728951109,386314552438,1398132674955,5059626441177,18308871648576,66249898660801
%N A208736 Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank level between 0 and 1.
%C A208736 Uniform used in the sense of Retakh, Serconek and Wilson. We use Stanley's definition of graded poset: all maximal chains have the same length n (which also implies all maximal elements have maximal rank.)
%D A208736 R. Stanley, Enumerative combinatorics. Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.
%H A208736 Vincenzo Librandi, <a href="/A208736/b208736.txt">Table of n, a(n) for n = 0..1000</a>
%H A208736 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 A208736 Wikipedia, <a href="http://en.wikipedia.org/wiki/Graded_poset">Graded poset</a>
%H A208736 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-21,20,-5).
%F A208736 a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4), a(2) = 0, a(3) = 1, a(4) = 5, a(5) = 22.
%F A208736 G.f.: (x^3 - 3*x^4 + 3*x^5)/(1 - 8*x + 21*x^2 - 20*x^3 + 5*x^4); (x^3 * (1 - 3*x + 3*x^2))/((1 - 3*x + x^2)*(1 - 5*x + 5*x^2)) .
%F A208736 a(n) = A081567(n-2) - A001519(n-1).
%t A208736 Join[{0, 0}, LinearRecurrence[{8, -21, 20, -5}, {0, 1, 5, 22}, 40]]
%o A208736 (Python)
%o A208736 def a(n, d={0:0,1:0,2:0,3:1,4:5,5:22}):
%o A208736     if n in d:
%o A208736         return d[n]
%o A208736     d[n]=8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4)
%o A208736     return d[n]
%Y A208736 Cf. A208737, A206901, A206902, A206947-A206950, A001906, A025192, A081567, A124302, A124292, A088305, A086405, A012781.
%K A208736 nonn,easy
%O A208736 0,5
%A A208736 _David Nacin_, Mar 01 2012