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 A032263 Number of ways to partition n labeled elements into 4 pie slices allowing the pie to be turned over; number of 2-element proper antichains of an n-element set. 24
 0, 0, 0, 3, 30, 195, 1050, 5103, 23310, 102315, 437250, 1834503, 7597590, 31175235, 127067850, 515396703, 2083011870, 8396420955, 33779000850, 135696347703, 544527210150, 2183335871475, 8749027724250, 35043169903503, 140313869216430, 561679070838795 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A000453. Sequence in context: A177727 A132413 A344907 * A003771 A121100 A203366 Adjacent sequences: A032260 A032261 A032262 * A032264 A032265 A032266 KEYWORD nonn,easy,nice AUTHOR Christian G. Bower EXTENSIONS Alternative description from Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic More terms from Vincenzo Librandi, Oct 19 2013 STATUS approved

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