%I #30 Mar 08 2023 13:33:41

%S 0,2,12,42,112,252,504,924,1584,2574,4004,6006,8736,12376,17136,23256,

%T 31008,40698,52668,67298,85008,106260,131560,161460,196560,237510,

%U 285012,339822,402752,474672,556512,649264,753984,871794,1003884,1151514,1316016,1498796

%N Number of ways 2*n-1 people can vote on three candidates so that the Condorcet paradox arises.

%H G. C. Greubel, <a href="/A277935/b277935.txt">Table of n, a(n) for n = 1..1000</a>

%H R. Embar, D. Zeilberger,<a href="https://doi.org/10.54550/ECA2022V2S3R22">Counting Condorcet</a>, Enum. Combin. Applic. 2 (2022) #S2R22

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Condorcet_paradox">Condorcet paradox</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F a(n) = (2/5!)*n*(n-1)*(n+3)*(n+2)*(n+1).

%F From _N. J. A. Sloane_, Nov 10 2016: (Start)

%F a(n) = 2*binomial(n+3,5) = 2*A000389(n+3).

%F G.f.: 2*x^2/(1-x)^6. (End)

%F E.g.f.: x^2*(60 + 60*x + 15*x^2 + x^3)*exp(x)/60. - _G. C. Greubel_, Nov 25 2017

%e For n=2 (three voters), the two possible ways the Condorcet paradox arises are:

%e 1) one voter prefers A to B to C, one prefers B to C to A, and one prefers C to A to B.

%e 2) one voter prefers A to C to B, one prefers C to B to A, and one prefers B to A to C.

%t Table[(2/5!)*n*(n - 1)*(n + 3)*(n + 2)*(n + 1), {n, 1, 50}] (* _G. C. Greubel_, Nov 25 2017 *)

%t a[n_] := 2 Binomial[n + 3, 5]; Array[a, 40] (* or *)

%t Rest@ CoefficientList[ Series[2 x^2/(x - 1)^6, {x, 0, 40}], x] (* or *)

%t Range[0, 40]! CoefficientList[ Series[x^2 (x^3 + 15x^2 + 60x + 60) Exp[x]/60, {x, 0, 40}], x] (* or *)

%t LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 2, 12, 42, 112, 252, 504}, 40] (* _Robert G. Wilson v_, Nov 25 2017 *)

%o (PARI) for(n=1,30, print1((2/5!)*n*(n-1)*(n+3)*(n+2)*(n+1), ", ")) \\ _G. C. Greubel_, Nov 25 2017

%o (Magma) [(2/Factorial(5))*n*(n-1)*(n+3)*(n+2)*(n+1): n in [1..30]]; // _G. C. Greubel_, Nov 25 2017

%Y Cf. A000389.

%K nonn,easy

%O 1,2

%A _Andrew Lohr_, Nov 04 2016