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%I #46 Aug 04 2022 06:04:27
%S 0,1,4,13,28,1199,239,113149,1487,14345663,292223,17110600987,
%T 14849671,545242142639,961780559,1704615588759647,856088316689,
%U 7836371329207844977,1103759659545457,16895087931630048788047,59954362566895631,2699144613568894213138579,28918424475964028179
%N Random walk in R^3: Numerators of the expected distance after n steps.
%C 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 numerators.
%C 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 numerators.
%H Ludovic Schwob, <a href="/A340002/b340002.txt">Table of n, a(n) for n = 0..99</a>
%F a(n)/A340003(n) ~ 2*sqrt(2*n)/sqrt(3*Pi).
%F a(n)/A340003(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
%e 0, 1, 4/3, 13/8, 28/15, 1199/576, 239/105, 113149/46080, 1487/567, 14345663/5160960, ... = A340002/A340003.
%Y See A340003 for denominators.
%K nonn,frac
%O 0,3
%A _Ludovic Schwob_, Dec 26 2020
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