A206947 - OEIS

%I #38 May 17 2021 08:49:14

%S 0,0,0,2,14,70,306,1248,4888,18666,70110,260414,959882,3519232,

%T 12854064,46824210,170243566,618125238,2242100898,8126927456,

%U 29442587720,106626616954,386046638142,1397431266222,5057790129274,18304064121600,66237312391776

%N Number of nonisomorphic graded posets with 0 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank above 0.

%C Here, the term uniform 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 R. Stanley, Enumerative combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

%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_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-21,20,-5).

%F a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4), a(1)=0, a(2)=0, a(3)=2, a(4)=14.

%F G.f.: (2*(1-x)*x^3)/((1-3*x+x^2)*(1-5*x+5*x^2)).

%F a(n) = A081567(n-1) - A001906(n).

%t Join[{0}, LinearRecurrence[{8, -21, 20, -5}, {0, 0, 2, 14}, 40]]

%o (Python)

%o def a(n,adict={0:0,1:0,2:0,3:2,4:14}):

%o     if n in adict:

%o         return adict[n]

%o     adict[n]=8*a(n-1)-21*a(n-2)+20*a(n-3)-5*a(n-4)

%o     return adict[n]

%Y Cf. A206948 (removing unique maximal element.)

%Y Cf. A206949, A206950 (allowing one or two elements in each rank level above 0 with and without maximal element.)

%K nonn,easy

%O 0,4

%A _David Nacin_, Feb 13 2012