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A234600 Denominators of the expectation of the process defined by randomly moving 2n balls between bins. 1

%I #14 Apr 29 2014 20:02:15

%S 1,3,5,105,63,1155,6435,45045,85085,2909907,1322685,111546435,

%T 128707425,717084225,9704539845,4512611027925,265447707525,

%U 501401225325,8787716212275,33393321606645

%N Denominators of the expectation of the process defined by randomly moving 2n balls between bins.

%C Start with two bins, one empty and the other containing 2n balls, n >= 1. Each turn, randomly select a ball and move it from the bin it is in to the other bin. Continue until both bins contain the same number of balls. The numbers in this sequence are the denominators of the rational expressions for the expected number of moves required to end up with the same number of balls in each bin.

%o (Maxima)

%o E(n) := (

%o block (

%o [T, P, S, i, t],

%o T[0] : 1,

%o T[1] : 1,

%o for i : 2 thru n do (

%o T[i] : T[i-1] - (n+i)*(n-i+1)*T[i-2]/(4*n^2)),

%o P[n] : 1,

%o for i : n - 1 step -1 thru 1 do (

%o P[i] : (n+i+1)*P[i+1]/(2*n)),

%o S : 0,

%o for i : 1 thru n do (

%o S : S + P[i]*T[i-1]/T[n]),

%o disp(S)

%o )

%o )$

%Y Cf. A233470 (numerators).

%K nonn,frac

%O 1,2

%A _Aaron Clark_ and Stephen Gueble, Apr 19 2014

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)