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A353194 Number of Condorcet voting profiles with three candidates and 2n-1 voters. 1
0, 12, 540, 21000, 785820, 28956312, 1058809752, 38545567632, 1399354322652, 50707958458872, 1835099465988360, 66348521294296176, 2397139928161319640, 86559958069097395440, 3124302168622853150640, 112729791393354644416800 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
All terms are multiples of 12.
LINKS
Shalosh B. Ekhad, More terms.
Rebecca Embar and Doron Zeilberger, Counting Condorcet.
Doron Zeilberger, Condorcet3 Maple package.
FORMULA
a(n) = (4*(19*n^2-57*n+45)/(n-1)^2)*a(n-1) - (36*(2*n-3)*(22*n^2-99*n+111)/((n-2)*(n-1)^2))*a(n-2) + (1296*(n-3)*(2*n-3)*(2*n-5)/((n-2)*(n-1)^2))*a(n-3).
a(n) = 2*(Sum_{i1=0..n-2} Sum_{i2=0..n-2-i1} Sum_{i3=0..n-2-i1-i2} Sum_{i4=0..n-2-i1-i2-i3} Sum_{i5=0..n-2-i1-i2-i3-i4} ((2*n-1)!/((n-1-i2-i3-i5)!*i2!*i3!*(i2+i4+i5+1)!*(n-1-i1-i2-i4)!*i1!))).
a(n) ~ (1/4 - 3*arcsin(1/3)/(2*Pi)) * 6^(2*n - 1) [Guilbaud, 1952].
MAPLE
#(From Maple package Condorcet3 by Doron Zeilberger)
#NuCo(N):The first N terms of the sequence "number of Condorcet vote-profiles" with 2v-1 voters and three candidates. Using the third-order recurrence. Try:
#NuCo(100);
NuCo:=proc(N) local L, n, kha: L:=[0, 12, 540]: if N<=3 then RETURN(L[N]): fi: for n from 4 to N do kha:=4*(19*n^2-57*n+45)/(n-1)^2*L[-1]-36*(2*n-3)*(22*n^2-99*n+111)/(n-2)/(n-1)^2*L[-2]+1296*(n-3)*(2*n-3)*(2*n-5)/(n-2)/(n-1)^2*L[-3]: L:=[L[2], L[3], kha]: od: L[-1]: end: seq(NuCo(n), n=1..16);
MATHEMATICA
RecurrenceTable[{a[n] == (4*(19*n^2 - 57*n + 45)/(n-1)^2) * a[n-1] - (36*(2*n - 3)*(22*n^2 - 99*n + 111)/((n-2)*(n-1)^2)) * a[n-2] + (1296*(n-3)*(2*n - 3)*(2*n - 5)/((n-2)*(n-1)^2)) * a[n-3], a[1] == 0, a[2] == 12, a[3] == 540}, a[n], {n, 1, 20}] (* Vaclav Kotesovec, May 20 2022 *)
PROG
(PARI) a(n) = 2*sum(i1=0, n-2, sum(i2=0, n-2-i1, sum(i3=0, n-2-i1-i2, sum(i4=0, n-2-i1-i2-i3, sum(i5=0, n-2-i1-i2-i3-i4, ((2*n-1)!/((n-1-i2-i3-i5)!*i2!*i3!*(i2+i4+i5+1)!*(n-1-i1-i2-i4)!*i1!))))))) \\ Michel Marcus, May 03 2022
CROSSREFS
Sequence in context: A281030 A282589 A067733 * A285748 A184423 A064344
KEYWORD
nonn
AUTHOR
Rebecca Embar, Apr 29 2022
STATUS
approved

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Last modified July 17 09:39 EDT 2024. Contains 374363 sequences. (Running on oeis4.)