A340003 Random walk in R^3: Denominators of the expected distance after n steps.

1, 1, 3, 8, 15, 576, 105, 46080, 567, 5160960, 99792, 5573836800, 4633200, 163499212800, 277992000, 476109707673600, 231567336000, 2056793937149952000, 281585880576000, 4195859631785902080000, 14514472207872000, 637770664031457116160000, 6676657215621120000

Offset

0,3

Comments

The random variables X_n are defined by X_0 = 0 and X_(n+1) = X_n + U_n where U_n are i.i.d. random variables with uniform distribution on the 2-dimensional sphere. Then a(n) = E(|X_n|), take denominators.

Let (V_n)_n be i.i.d. random variables with uniform distribution on the interval [-2,2]. Then a(n) = E(|V_1+...+V_n|), take denominators.

Links

Ludovic Schwob, Table of n, a(n) for n = 0..99

Formula

A340002(n)/a(n) ~ 2*sqrt(2*n)/sqrt(3*Pi).

A340002(n)/a(n) = (1/(2^(n-2) * (n+1)!)) * Sum_{k=0..floor((n-1)/2)} (-1)^k * C(n,k) * (n-2*k)^(n+1). - Ludovic Schwob, Jun 11 2022

Crossrefs

See A340002 for numerators.

Sequence in context: A325904 A362273 A094357 * A136532 A368041 A030417

Adjacent sequences: A340000 A340001 A340002 * A340004 A340005 A340006

Keyword

nonn,frac

Author

Ludovic Schwob, Dec 26 2020

Status

approved