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A277935 Number of ways 2*n-1 people can vote on three candidates so that the Condorcet paradox arises. 2
0, 2, 12, 42, 112, 252, 504, 924, 1584, 2574, 4004, 6006, 8736, 12376, 17136, 23256, 31008, 40698, 52668, 67298, 85008, 106260, 131560, 161460, 196560, 237510, 285012, 339822, 402752, 474672, 556512, 649264, 753984, 871794, 1003884, 1151514, 1316016, 1498796 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

Wikipedia, Condorcet paradox

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

FORMULA

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

From N. J. A. Sloane, Nov 10 2016: (Start)

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

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

E.g.f.: x^2*(60 + 60*x + 15*x^2 + x^3)*exp(x)/60. - G. C. Greubel, Nov 25 2017

EXAMPLE

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

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

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

MATHEMATICA

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

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

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

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

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

PROG

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

(MAGMA) [(2/Factorial(5))*n*(n-1)*(n+3)*(n+2)*(n+1): n in [1..30]]; // G. C. Greubel, Nov 25 2017

CROSSREFS

Cf. A000389.

Sequence in context: A048014 A094702 A001621 * A290928 A232584 A189491

Adjacent sequences:  A277932 A277933 A277934 * A277936 A277937 A277938

KEYWORD

nonn,easy

AUTHOR

Andrew Lohr, Nov 04 2016

STATUS

approved

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Last modified February 21 06:43 EST 2018. Contains 299390 sequences. (Running on oeis4.)