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%I #28 Jul 17 2019 08:36:32
%S 1,8,23,704,563,13016,88069,728576,1593269,62075752,31730711,
%T 2977423552,3788707301,23104065256,340028535787,170983243313152,
%U 10823198495797,21904260478904,409741429887649,1656090499861696
%N Numerators 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. On each turn, randomly select a ball and move it from its current bin to the other bin. Continue until each bin contains the same number of balls. The numbers in this sequence are the numerators of the rational expressions for the expected number of moves required to end up with the same number of balls in each bin.
%C From _Jon E. Schoenfield_, May 02 2014, updated Jul 16 2019: (Start)
%C Let E(n) = A233470(n) / A234600(n) be the expected number of moves required when the total number of balls is 2n. As n increases, it appears that E(n) asymptotically approaches
%C (n/2)*log(n) + c_0*n + c_1/n + c_2/n^3 + c_3/n^5 + ... + c_j / n^(2j-1) + ...
%C where
%C c_0 = log(2) + gamma/2 (where gamma is the Euler-Mascheroni constant; cf. A001620),
%C c_j = B(2j) * (2^(2j-1)-1) / (j*2^(2j+1)) for j > 0, and
%C B(2j) is the (2j)-th Bernoulli number.
%C (Thanks to Jean-Marc Luck for identifying c_0 as log(2) + gamma/2.) (End)
%e 1/1, 8/3, 23/5, 704/105, 563/63, 13016/1155, 88069/6435, 728576/45045, 1593269/85085, 62075752/2909907, ... = A233470/A234600
%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. A001620, A234600 (denominators).
%K nonn,frac
%O 1,2
%A _Aaron Clark_ and Stephen Gueble, Apr 19 2014
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