A253095 Moments of 4-step random walk in 4 dimensions.

1, 4, 22, 148, 1144, 9784, 90346, 885868, 9115276, 97578688, 1079676448, 12285725632, 143204046496, 1704422018992, 20660609113186, 254522834851516, 3180935346538684, 40269426101933392, 515743456513546072, 6675036087017279056, 87221496402779437696, 1149701868292524559744

Offset: 0

Author

N. J. A. Sloane, Feb 16 2015

Keywords

• nonn

Links

• J. M. Borwein, A short walk can be beautiful, Journal of Humanistic Mathematics, Volume 6 Issue 1 (January 2016), pages 86-109.
• J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
• J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
• Jonathan M. Borwein, Armin Straub and Christophe Vignat, Densities of short uniform random walks, Part II: Higher dimensions, Preprint, 2015.

Programs

Maple

W := proc(n,nu,twok)
    option remember;
    local k;
    k := twok/2 ;
    if n = 2 and nu = 1 then
        binomial(2*k+2,k+1)/(k+2) ;
    else
        add( procname(n-1,nu,2*j)*binomial(k,j)*(k+nu)!*nu!/(k-j+nu)!/(j+nu)!,j=0..k) ;
        simplify(%,GAMMA) ;
    end if;
end proc:
A253095 := proc(n)
    W(4,1,n) ;
end proc:
seq(A253095(2*n),n=0..25) ; # R. J. Mathar, Jun 14 2015

Mathematica

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}]]];
A253095[n_] := W[4, 1, n];
Table[A253095[2n], {n, 0, 25}] (* Jean-François Alcover, Apr 16 2023, after R. J. Mathar *)