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%I #25 Apr 05 2024 10:10:43
%S 1,4,22,148,1144,9784,90346,885868,9115276,97578688,1079676448,
%T 12285725632,143204046496,1704422018992,20660609113186,
%U 254522834851516,3180935346538684,40269426101933392,515743456513546072,6675036087017279056,87221496402779437696,1149701868292524559744
%N Moments of 4-step random walk in 4 dimensions.
%H J. M. Borwein, <a href="https://scholarship.claremont.edu/jhm/vol6/iss1/7">A short walk can be beautiful</a>, Journal of Humanistic Mathematics, Volume 6 Issue 1 (January 2016), pages 86-109.
%H J. M. Borwein, <a href="https://carmamaths.org/resources/jon/OEIStalk.pdf">Adventures with the OEIS: Five sequences Tony may like</a>, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
%H J. M. Borwein, <a href="/A060997/a060997.pdf">Adventures with the OEIS: Five sequences Tony may like</a>, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
%H Jonathan M. Borwein, Armin Straub and Christophe Vignat, <a href="https://arminstraub.com/pub/dwalks">Densities of short uniform random walks, Part II: Higher dimensions</a>, Preprint, 2015.
%p W := proc(n,nu,twok)
%p option remember;
%p local k;
%p k := twok/2 ;
%p if n = 2 and nu = 1 then
%p binomial(2*k+2,k+1)/(k+2) ;
%p else
%p add( procname(n-1,nu,2*j)*binomial(k,j)*(k+nu)!*nu!/(k-j+nu)!/(j+nu)!,j=0..k) ;
%p simplify(%,GAMMA) ;
%p end if;
%p end proc:
%p A253095 := proc(n)
%p W(4,1,n) ;
%p end proc:
%p seq(A253095(2*n),n=0..25) ; # _R. J. Mathar_, Jun 14 2015
%t W[n_, nu_, twok_] := W[n, nu, twok] = Module[{k}, k = twok/2; If[n == 2 && nu == 1, Binomial[2k+2, k+1]/(k+2), Sum[W[n-1, nu, 2j]*Binomial[k, j]*(k+nu)!*nu!/(k-j+nu)!/(j+nu)!, {j, 0, k}]]];
%t A253095[n_] := W[4, 1, n];
%t Table[A253095[2n], {n, 0, 25}] (* _Jean-François Alcover_, Apr 16 2023, after _R. J. Mathar_ *)
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Feb 16 2015
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