OFFSET
1,4
COMMENTS
A proper antichain is an antichain iff each two of its members have a nonempty intersection.
Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements {x,y} of P(A) for which x and y are intersecting but for which x is not a subset of y and y is not a subset of x. This is just a different formulation of the alternative sequence description. - Ross La Haye, Jan 09 2008
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. G. Bower, Transforms (2)
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).
FORMULA
"DIJ[ 4 ]" (bracelet, indistinct, labeled, 4 parts) transform of 1, 1, 1, 1, ...
3*S(n,4) = (4^n-4*3^n+6*2^n-4)/8. - R. J. Mathar, Feb 26 2008
G.f.: 3*x^4/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)). - Colin Barker, May 29 2012
a(n) = 3*A000453(n). - Alois P. Heinz, Jan 24 2018
E.g.f.: (exp(x) - 1)^4/8. - Stefano Spezia, Apr 06 2022
MAPLE
A032263 := proc(n) (4^n-4*3^n+6*2^n-4)/8 ; end: seq(A032263(n), n=1..20) ; # R. J. Mathar, Feb 26 2008
MATHEMATICA
CoefficientList[Series[(3x^4)/((1-x)(1-2x)(1-3x)(1-4x)), {x, 0, 40}], x] (* Harvey P. Dale, Feb 28 2013 *)
PROG
(Magma) I:=[0, 0, 0, 3]; [n le 4 select I[n] else 10*Self(n-1)-35*Self(n-2)+50*Self(n-3)-24*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Oct 19 2013
(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -24, 50, -35, 10]^(n-1)*[0; 0; 0; 3])[1, 1] \\ Charles R Greathouse IV, Feb 09 2017
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Alternative description from Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic
More terms from Vincenzo Librandi, Oct 19 2013
STATUS
approved