A080687 Number of labeled n-element posets with no 3-element antichain.

1, 1, 3, 18, 174, 2370, 41850, 908460, 23393160, 696752280, 23558056200, 891259815600, 37298874135600, 1710662148795600, 85319825069278800, 4597474487169564000, 266164417718126928000, 16475817276720193392000

Offset: 0

Author

Detlef Pauly (dettodet(AT)yahoo.de), Mar 03 2003

Keywords

• nonn

Links

• Graham Brightwell and Sarah Goodall, The number of partial orders of fixed width, Order, 13 (1996), 315-337.

Crossrefs

Cf. A006251 for the unlabeled analog.

Programs

Maple

A080687 := proc(n)
    (3-2*x-sqrt(1-4*x)) / (2-2*x+x^2) ;
    coeftayl(%,x=0,n) ;
    %*n! ;
end proc:
seq(A080687(n),n=0..30) ; # R. J. Mathar, Feb 08 2021

Mathematica

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 *)
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 *)

Formula

E.g.f.: (3-2*x-sqrt(1-4*x)) / (2-2*x+x^2).

a(n) ~ n^(n-1)*2^(2*n+7/2)/(25*exp(n)). - Vaclav Kotesovec, Sep 29 2013

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

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