A233470 Numerators of the expectation of the process defined by randomly moving 2n balls between bins.

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

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.

From Jon E. Schoenfield, May 02 2014, updated Jul 16 2019: (Start)

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^(2j-1) + ...

where

c_0 = log(2) + gamma/2 (where gamma is the Euler-Mascheroni constant; cf. A001620),

c_j = B(2j) * (2^(2j-1)-1) / (j*2^(2j+1)) for j > 0, and

B(2j) is the (2j)-th Bernoulli number.

(Thanks to Jean-Marc Luck for identifying c_0 as log(2) + gamma/2.) (End)

Links

Table of n, a(n) for n=1..20.

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[i-1] - (n+i)*(n-i+1)*T[i-2]/(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[i-1]/T[n]),
        disp(S)
    )
)$

Crossrefs

Cf. A001620, A234600 (denominators).

Sequence in context: A304414 A305949 A220973 * A334720 A172390 A250035

Adjacent sequences: A233467 A233468 A233469 * A233471 A233472 A233473

Keyword

nonn,frac

Author

Aaron Clark and Stephen Gueble, Apr 19 2014

Status

approved