This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A277935 #30 Mar 08 2023 13:33:41
%S A277935 0,2,12,42,112,252,504,924,1584,2574,4004,6006,8736,12376,17136,23256,
%T A277935 31008,40698,52668,67298,85008,106260,131560,161460,196560,237510,
%U A277935 285012,339822,402752,474672,556512,649264,753984,871794,1003884,1151514,1316016,1498796
%N A277935 Number of ways 2*n-1 people can vote on three candidates so that the Condorcet paradox arises.
%H A277935 G. C. Greubel, <a href="/A277935/b277935.txt">Table of n, a(n) for n = 1..1000</a>
%H A277935 R. Embar, D. Zeilberger,<a href="https://doi.org/10.54550/ECA2022V2S3R22">Counting Condorcet</a>, Enum. Combin. Applic. 2 (2022) #S2R22
%H A277935 Wikipedia, <a href="https://en.wikipedia.org/wiki/Condorcet_paradox">Condorcet paradox</a>
%H A277935 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A277935 a(n) = (2/5!)*n*(n-1)*(n+3)*(n+2)*(n+1).
%F A277935 From _N. J. A. Sloane_, Nov 10 2016: (Start)
%F A277935 a(n) = 2*binomial(n+3,5) = 2*A000389(n+3).
%F A277935 G.f.: 2*x^2/(1-x)^6. (End)
%F A277935 E.g.f.: x^2*(60 + 60*x + 15*x^2 + x^3)*exp(x)/60. - _G. C. Greubel_, Nov 25 2017
%e A277935 For n=2 (three voters), the two possible ways the Condorcet paradox arises are:
%e A277935 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 A277935 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 A277935 Table[(2/5!)*n*(n - 1)*(n + 3)*(n + 2)*(n + 1), {n, 1, 50}] (* _G. C. Greubel_, Nov 25 2017 *)
%t A277935 a[n_] := 2 Binomial[n + 3, 5]; Array[a, 40] (* or *)
%t A277935 Rest@ CoefficientList[ Series[2 x^2/(x - 1)^6, {x, 0, 40}], x] (* or *)
%t A277935 Range[0, 40]! CoefficientList[ Series[x^2 (x^3 + 15x^2 + 60x + 60) Exp[x]/60, {x, 0, 40}], x] (* or *)
%t A277935 LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 2, 12, 42, 112, 252, 504}, 40] (* _Robert G. Wilson v_, Nov 25 2017 *)
%o A277935 (PARI) for(n=1,30, print1((2/5!)*n*(n-1)*(n+3)*(n+2)*(n+1), ", ")) \\ _G. C. Greubel_, Nov 25 2017
%o A277935 (Magma) [(2/Factorial(5))*n*(n-1)*(n+3)*(n+2)*(n+1): n in [1..30]]; // _G. C. Greubel_, Nov 25 2017
%Y A277935 Cf. A000389.
%K A277935 nonn,easy
%O A277935 1,2
%A A277935 _Andrew Lohr_, Nov 04 2016