A387183 Denominators of the expected number of steps to hit the opposite corner by simple random walk on the n-cube.

1, 1, 1, 3, 3, 5, 15, 105, 35, 63, 315, 1155, 3465, 6435, 3003, 9009, 9009, 17017, 153153, 2909907, 692835, 1322685, 14549535, 111546435, 66927861, 128707425, 185910725, 717084225, 5019589575, 9704539845, 145568097675, 4512611027925, 136745788725, 265447707525

Offset

1,4

Links

Table of n, a(n) for n=1..34.

Wikipedia, Ehrenfest model

Wikipedia, Hypercube

Formula

a(n) = denominator(2^(n-1) * Sum_{i=0..n-1} 1/binomial(n-1,i)).

Example

1, 4, 10, 64/3, 128/3, 416/5, 2416/15, 32768/105, 21248/35, 74752/63, ...

Maple

t := n -> 2^(n-1)*add(1/binomial(n-1, i), i=0..n-1):
a := n -> denom(t(n)):
seq(a(n), n=1..34);

Mathematica

t[n_] := 2^(n - 1) Sum[1/Binomial[n - 1, i], {i, 0, n - 1}] // Together;
a[n_] := Denominator[t[n]];
Array[a, 34]

Prog

(MATLAB)
function q = cube_hit_den(N)
    q = sym(zeros(1, N));
    for n = 1:N
        t = sym(0);
        for i = 0:n-1
            t = t + 1/sym(nchoosek(n-1, i));
        end
        t = t * sym(2)^(n-1);
        [~, qn] = numden(t);
        q(n) = qn;
    end
end

Crossrefs

Cf. A384229 (numerators).

Sequence in context: A069834 A064038 A051684 * A209388 A195583 A370973

Adjacent sequences: A387180 A387181 A387182 * A387184 A387185 A387186

Keyword

nonn,frac

Author

Joost de Winter, Aug 20 2025

Status

approved