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%I #20 Feb 08 2021 06:22:50
%S 1,1,3,18,174,2370,41850,908460,23393160,696752280,23558056200,
%T 891259815600,37298874135600,1710662148795600,85319825069278800,
%U 4597474487169564000,266164417718126928000,16475817276720193392000
%N Number of labeled n-element posets with no 3-element antichain.
%H Graham Brightwell and Sarah Goodall, <a href="http://dx.doi.org/10.1007/BF00405592">The number of partial orders of fixed width</a>, Order, 13 (1996), 315-337.
%F E.g.f.: (3-2*x-sqrt(1-4*x)) / (2-2*x+x^2).
%F a(n) ~ n^(n-1)*2^(2*n+7/2)/(25*exp(n)). - _Vaclav Kotesovec_, Sep 29 2013
%F a(n) = 2^(-(n+2))*n!*((3-i)*(1-i)^n + (3+i)*(1+i)^n - (1+i)*(-8)^n*binomial(1/2,n)*(2F1(1,-n; 3/2 - n; (1-i)/8) - i*2F1(1, -n; 3/2 - n; (1+i)/8))). - _Benedict W. J. Irwin_, May 27 2016
%F D-finite with recurrence 2*a(n) +2*(-5*n+6)*a(n-1) +3*(n-1)*(3*n-4)*a(n-2) -2*(n-1)*(n-2)*(2*n-3)*a(n-3)=0. - _R. J. Mathar_, Feb 08 2021
%p A080687 := proc(n)
%p (3-2*x-sqrt(1-4*x)) / (2-2*x+x^2) ;
%p coeftayl(%,x=0,n) ;
%p %*n! ;
%p end proc:
%p seq(A080687(n),n=0..30) ; # _R. J. Mathar_, Feb 08 2021
%t CoefficientList[Series[(3-2*x-Sqrt[1-4*x])/(2-2*x+x^2), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 29 2013 *)
%t Table[2^(-(n + 2)) n! ((3 - I) (1 - I)^n + (3 + I) (1 + I)^n - (1 + I) (-8)^n Binomial[1/2,n] (Hypergeometric2F1[1, -n, 3/2 - n, (1 - I)/8] - I*Hypergeometric2F1[1, -n, 3/2 - n, (1 + I)/8])), {n, 0, 10}] (* _Benedict W. J. Irwin_, May 27 2016 *)
%Y Cf. A006251 for the unlabeled analog.
%K nonn
%O 0,3
%A Detlef Pauly (dettodet(AT)yahoo.de), Mar 03 2003
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