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%I #32 May 21 2021 08:09:47
%S 0,0,0,1,7,37,175,778,3325,13837,56524,227866,909832,3607294,14227447,
%T 55894252,218937532,855650749,3338323915,13007422705,50631143323,
%U 196928737582,765495534433,2974251390529,11552064922624,44856304154086
%N Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with no 3-element antichain.
%C 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 R. Stanley, Enumerative combinatorics. Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.
%H Vincenzo Librandi, <a href="/A208737/b208737.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 Wikipedia, <a href="http://en.wikipedia.org/wiki/Graded_poset">Graded poset</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (10,-36,57,-39,9).
%F a(n) = 10*a(n-1) - 36*a(n-2) + 57*a(n-3) - 39*a(n-4) + 9*a(n-5), a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 7, a(5) = 37.
%F G.f: (x^3 - 3*x^4 + 3*x^5)/(1 - 10*x + 36*x^2 - 57*x^3 + 39*x^4 - 9*x^5); (x^3*(1 - 3*x + 3*x^2)) / ((1 - x) (1 - 3*x) (1 - 6*x + 9*x^2 - 3*x^3)).
%F a(n) = A124292(n) - A124302(n).
%t Join[{0}, LinearRecurrence[{10, -36, 57, -39, 9}, {0, 0, 1, 7, 37}, 40]]
%o (Python)
%o def a(n, d={0:0,1:0,2:0,3:1,4:7,5:37}):
%o if n in d:
%o return d[n]
%o d[n]=10*a(n-1) - 36*a(n-2) + 57*a(n-3) - 39*a(n-4) + 9*a(n-5)
%o return d[n]
%Y Cf. A208736, A206901, A206902, A206947-A206950, A001906, A025192, A081567, A124302, A124292, A088305, A086405, A012781.
%K nonn,easy
%O 0,5
%A _David Nacin_, Mar 01 2012
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