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%I #41 Sep 08 2022 08:46:01
%S 1,2,8,36,166,768,3554,16446,76102,352152,1629536,7540458,34892452,
%T 161460114,747134894,3457265922,15998031616,74028732924,342557973998,
%U 1585140808368,7335025230994,33941839649382,157061283704438,726779900373936,3363075935260696
%N Number of nonisomorphic graded posets with 0 and uniform Hasse diagram of rank n with no 3-element antichain.
%C 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.
%C Uniform (in the definition) used in the sense of Retakh, Serconek and Wilson (see paper in Links lines). - _David Nacin_, Mar 01 2012
%H Vincenzo Librandi, <a href="/A206902/b206902.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_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-7,3).
%F a(n) = 6*a(n-1) - 7*a(n-2) + 3*a(n-3), a(1)=2, a(2)=8, a(3)=36.
%F G.f.: (1 -4*x +3*x^2 -x^3)/(1 -6*x +7*x^2 -3*x^3).
%t LinearRecurrence[{6,-7,3}, {1,2,8,36}, 30] (* _Vincenzo Librandi_, Feb 27 2012 *)
%o (Python)
%o def a(n, adict={1:2,2:8,3:36}):
%o if n in adict:
%o return adict[n]
%o adict[n]=6*a(n-1)-7*a(n-2)+3*a(n-3)
%o return adict[n]
%o (PARI) my(x='x+O('x^30)); Vec((1-4*x+3*x^2-x^3)/(1-6*x+7*x^2-3*x^3)) \\ _G. C. Greubel_, May 21 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-4*x +3*x^2-x^3)/(1-6*x+7*x^2-3*x^3) )); // _G. C. Greubel_, May 21 2019
%o (Sage) ((1-4*x+3*x^2-x^3)/(1-6*x+7*x^2-3*x^3)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 21 2019
%o (GAP) a:=[2,8,36];; for n in [4..30] do a[n]:=6*a[n-1]-7*a[n-2]+3*a[n-3]; od; Concatenation([1], a); # _G. C. Greubel_, May 21 2019
%Y Cf. A025192 (adding a unique maximal element).
%Y Cf. A124292, A206901 (dropping uniformity with and without maximal element).
%K nonn,easy
%O 0,2
%A _David Nacin_, Feb 13 2012