In Maple notation,
E.g.f. of r: ((135/8)*z+4)*cos((2/3)*arcsin((3/4)*sqrt(6)*sqrt(z)))/(1-(27/8)*z)^2+(3/8)*sqrt(z)*((135/4)*z+17)*sin((2/3)*arcsin((3/4)*sqrt(6)*sqrt(z)))*sqrt(6)/(1-(27/8)*z)^(5/2).
E.g.f. of r: 4*hypergeom([4/3,5/3],[1/2],27*z/8).
Integral representation as n-th moment of a signed function w(x) of bounded variation on (0,infinity),
w(x)=(8/3)*sqrt(3)*((1/36) *(128*x^2/81-40*x/3+20) *exp(-4*x/27) *BesselK(1/3,(4/27)*x)/Pi +(2/81)*x *(-5+16*x/9) *exp(-4*x/27) *BesselK(4/3,4*x/27)/Pi);
w(x)=-(14/243)*16^(2/3)*x^(2/3)*3 *hypergeom([13/6], [4/3], -(8/27)*x)/GAMMA(2/3)-(10/243)*sqrt(3)*16^(1/3)*x^(1/3)*9*GAMMA(2/3)*hypergeom([11/6], [2/3], -(8/27)*x)/Pi.
For x>3.32, w(x)>0.
w(0)=w(3.32)=limit(w(x),x=infinity)=0.
For x<3.32, w(x)<0.
r(n) = int(x^n*w(x), x=0..infinity), n>=0.
Asymptotics: r(n)->(1/1152)*sqrt(6)*(10368*n^2+10224*n+2161)*(27/8)^n*exp(-n)*(n^n), for n->infinity.
The rational values are given by 4*(-2*n+1)*r(n) + 3*(3*n+2)*(3*n+1) * r(n-1)=0. -
R. J. Mathar, Jul 20 2013
