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A253096 Moments of 5-step random walk in 4 dimensions.

%I A253096 #21 Apr 05 2024 11:10:14
%S A253096 1,5,35,305,3105,35505,444225,5970725,85068365,1272022745,19810304695,
%T A253096 319422093325,5307057746125,90508769121165,1579462112057595,
%U A253096 28130401719357645,510199561574590125,9405815167297415025,175977472926360962295,3336795660732459377085,64047222901288457886285
%N A253096 Moments of 5-step random walk in 4 dimensions.
%H A253096 J. M. Borwein, <a href="https://carmamaths.org/resources/jon/beauty.pdf">A short walk can be beautiful</a>, 2015.
%H A253096 Jonathan M. Borwein, Armin Straub and Christophe Vignat, <a href="http://carmamaths.org/resources/jon/dwalks.pdf">Densities of short uniform random walks, Part II: Higher dimensions</a>, Preprint, 2015.
%p A253096 W := proc(n,nu,twok)
%p A253096     option remember;
%p A253096     local k;
%p A253096     k := twok/2 ;
%p A253096     if n = 2 and nu = 1 then
%p A253096         binomial(2*k+2,k+1)/(k+2) ;
%p A253096     else
%p A253096         add( procname(n-1,nu,2*j)*binomial(k,j)*(k+nu)!*nu!/(k-j+nu)!/(j+nu)!,j=0..k) ;
%p A253096         simplify(%,GAMMA) ;
%p A253096     end if;
%p A253096 end proc:
%p A253096 A253096 := proc(n)
%p A253096     W(5,1,n) ;
%p A253096 end proc:
%p A253096 seq(A253096(2*n),n=0..25) ; # _R. J. Mathar_, Jun 14 2015
%Y A253096 Cf. A253095 (4-step), A253097 (6-step).
%K A253096 nonn
%O A253096 0,2
%A A253096 _N. J. A. Sloane_, Feb 16 2015
%E A253096 a(20) corrected by _Georg Fischer_, May 30 2022