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A353194 Number of Condorcet voting profiles with three candidates and 2n-1 voters.

%I A353194 #29 May 20 2022 03:30:17
%S A353194 0,12,540,21000,785820,28956312,1058809752,38545567632,1399354322652,
%T A353194 50707958458872,1835099465988360,66348521294296176,
%U A353194 2397139928161319640,86559958069097395440,3124302168622853150640,112729791393354644416800
%N A353194 Number of Condorcet voting profiles with three candidates and 2n-1 voters.
%C A353194 All terms are multiples of 12.
%H A353194 Shalosh B. Ekhad, <a href="https://sites.math.rutgers.edu/~zeilberg/tokhniot/oCondorcet3b.txt">More terms</a>.
%H A353194 Rebecca Embar and Doron Zeilberger, <a href="https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/cond3.pdf">Counting Condorcet</a>.
%H A353194 Doron Zeilberger, <a href="https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/cond3.html">Condorcet3 Maple package</a>.
%F A353194 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).
%F A353194 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!))).
%F A353194 a(n) ~ (1/4 - 3*arcsin(1/3)/(2*Pi)) * 6^(2*n - 1) [Guilbaud, 1952].
%p A353194 #(From Maple package Condorcet3 by Doron Zeilberger)
%p A353194 #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:
%p A353194 #NuCo(100);
%p A353194 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);
%t A353194 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 *)
%o A353194 (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
%Y A353194 Cf. A277935, A353230.
%K A353194 nonn
%O A353194 1,2
%A A353194 _Rebecca Embar_, Apr 29 2022