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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.

%I A032263 #44 Apr 07 2022 12:09:45
%S A032263 0,0,0,3,30,195,1050,5103,23310,102315,437250,1834503,7597590,
%T A032263 31175235,127067850,515396703,2083011870,8396420955,33779000850,
%U A032263 135696347703,544527210150,2183335871475,8749027724250,35043169903503,140313869216430,561679070838795
%N 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.
%C A032263 A proper antichain is an antichain iff each two of its members have a nonempty intersection.
%C A032263 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
%H A032263 Vincenzo Librandi, <a href="/A032263/b032263.txt">Table of n, a(n) for n = 1..1000</a>
%H A032263 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%H A032263 Ross La Haye, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/LaHaye/lahaye5.html">Binary Relations on the Power Set of an n-Element Set</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
%H A032263 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (10,-35,50,-24).
%F A032263 "DIJ[ 4 ]" (bracelet, indistinct, labeled, 4 parts) transform of 1, 1, 1, 1, ...
%F A032263 3*S(n,4) = (4^n-4*3^n+6*2^n-4)/8. - _R. J. Mathar_, Feb 26 2008
%F A032263 G.f.: 3*x^4/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)). - _Colin Barker_, May 29 2012
%F A032263 a(n) = 3*A000453(n). - _Alois P. Heinz_, Jan 24 2018
%F A032263 E.g.f.: (exp(x) - 1)^4/8. - _Stefano Spezia_, Apr 06 2022
%p A032263 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
%t A032263 CoefficientList[Series[(3x^4)/((1-x)(1-2x)(1-3x)(1-4x)),{x,0,40}],x] (* _Harvey P. Dale_, Feb 28 2013 *)
%o A032263 (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
%o A032263 (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
%Y A032263 Cf. A000453.
%K A032263 nonn,easy,nice
%O A032263 1,4
%A A032263 _Christian G. Bower_
%E A032263 Alternative description from _Vladeta Jovovic_, Goran Kilibarda, Zoran Maksimovic
%E A032263 More terms from _Vincenzo Librandi_, Oct 19 2013