A253094 - OEIS

%I #31 Nov 12 2023 12:08:27

%S 1,-5,6,2,6,18,66,278,1296,6528,34950,196578,1151610,6981102,43570170,

%T 278841438,1823991630,12162884778,82498605594,568140045918,

%U 3966323992074,28032955095198,200355706872054,1446628270673682,10542888272710224,77496225169484448

%N Related to residues of poles of moment function for random walks in 6 dimensions.

%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.

%F n*(n+2)*a(n) + 5*(-2*n^2+6*n-1)*a(n-1) + 9*(n-3)*(n-5)*a(n-2) = 0. - _R. J. Mathar_, Jun 14 2015

%p A253094 := proc(k)

%p     option remember;

%p     local nu,kno ;

%p     nu := 2;

%p     if k = -1 then

%p         0;

%p     elif k = 0 then

%p         1;

%p     else

%p         kno := k-1 ;

%p         procname(kno)/2*(20*(kno+1/2)^2-20*(kno+1/2)*nu-4*nu^2+1)-9*(kno-nu)*(kno-2*nu)*procname(kno-1) ;

%p         %/(kno+1)/(kno+nu+1) ;

%p     end if;

%p end proc:

%p seq(A253094(k),k=0..40) ; # _R. J. Mathar_, Jun 14 2015

%p ogf := (x-1)^4*hypergeom([1/3, 7/3],[3],-27*x*(x-1)^2/(9*x-1)^2)/(1-9*x)^(2/3);

%p series(ogf, x=0, 40); # _Mark van Hoeij_, Nov 12 2023

%t a[n_] := a[n] = Switch[n, 0, 1, 1, -5, _, (-9*n^2*a[n-2] + 10*n^2*a[n-1] + 72n*a[n-2] - 30n*a[n-1] - 135 a[n-2] + 5a[n-1])/(n(n+2))];

%t Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Apr 15 2023 *)

%K sign

%O 0,2

%A _N. J. A. Sloane_, Feb 16 2015