

A233470


Numerators of the expectation of the process defined by randomly moving 2n balls between bins.


2



1, 8, 23, 704, 563, 13016, 88069, 728576, 1593269, 62075752, 31730711, 2977423552, 3788707301, 23104065256, 340028535787, 170983243313152, 10823198495797, 21904260478904, 409741429887649, 1656090499861696
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OFFSET

1,2


COMMENTS

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.
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
(n/2)*log(n) + c_0*n + c_1/n + c_2/n^3 + c_3/n^5 + ... + c_j / n^(2j1) + ...
where
c_0 = log(2) + gamma/2 (where gamma is the EulerMascheroni constant; cf. A001620),
c_j = B(2j) * (2^(2j1)1) / (j*2^(2j+1)) for j > 0, and
B(2j) is the (2j)th Bernoulli number.
(Thanks to JeanMarc Luck for identifying c_0 as log(2) + gamma/2.) (End)


LINKS



EXAMPLE

1/1, 8/3, 23/5, 704/105, 563/63, 13016/1155, 88069/6435, 728576/45045, 1593269/85085, 62075752/2909907, ... = A233470/A234600


PROG

(Maxima)
E(n) := (
block (
[T, P, S, i, t],
T[0] : 1,
T[1] : 1,
for i : 2 thru n do (
T[i] : T[i1]  (n+i)*(ni+1)*T[i2]/(4*n^2)),
P[n] : 1,
for i : n  1 step 1 thru 1 do (
P[i] : (n+i+1)*P[i+1]/(2*n)),
S : 0,
for i : 1 thru n do (
S : S + P[i]*T[i1]/T[n]),
disp(S)
)
)$


CROSSREFS



KEYWORD

nonn,frac


AUTHOR



STATUS

approved



