OFFSET
0,1
COMMENTS
The first values with denominators > 1 occur at n = (43, 86, 87, 91, 107, 171, 172, ...). - G. C. Greubel, Jul 04 2017
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..355
FORMULA
In Maple notation,
E.g.f. of r(n): 2*hypergeom([3/4, 5/4, 3/2], [1/3, 2/3], (128/27)*z).
Integral representation as n-th moment of a signed function w(x) of bounded variation on (0,infinity),
w(x) = (5/64)*2^(3/4)*hypergeom([13/12, 17/12], [1/4, 1/2], -(27/128)*x)/(GAMMA(3/4)*x^(1/4))-(231/512)*2^(3/4)*GAMMA(3/4)*x^(1/4)*hypergeom([19/12, 23/12], [3/4, 3/2], -(27/128)*x)/Pi-(35/64)*sqrt(2)*sqrt(x)*hypergeom([11/6, 13/6], [5/4, 7/4], -(27/128)*x)/sqrt(Pi);
For x>5.723, w(x)>0.
w(0)=w(5.723)=limit(w(x),x=infinity)=0.
For x<5.723, w(x)<0.
r(n) = int(x^n*w(x), x=0..infinity), n>=0.
Asymptotics: r(n)-> (1/3888)*sqrt(3)*(41472*n^2+30816*n+4969)*(128/27)^n*exp(-n)*(n)^(n), for n->infinity.
3*(3*n-1)*(3*n-2)*r(n) -4*(4*n+1)*(2*n+1)*(4*n-1)*r(n-1)=0. - R. J. Mathar, Oct 08 2016
MAPLE
seq(numer(2*(4*n+2)!/((3*n)!*2^(n+1)), n=0..15)
MATHEMATICA
Table[(4 n + 2)!/((3 n)!*2^n), {n, 0, 15}] (* Michael De Vlieger, Oct 08 2016 *)(* generates values of r(n) *)
Table[Numerator[(4*n + 2)!/((3*n)! *2^n)], {n, 0, 50}] (* G. C. Greubel, Jul 04 2017 *)
PROG
(PARI) for(n=0, 50, print1(numerator((4*n + 2)!/((3*n)! *2^n)), ", ")) \\ G. C. Greubel, Jul 05 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Karol A. Penson, Jul 18 2013
STATUS
approved