OFFSET
0,2
COMMENTS
The denominators seem to coincide with A241756.
These are the partial sums of F. Morley's series Sum_{k>=0} (risefac(m,k)/k!)^3 for m=1/2, where risefac(x,k) = Product_{j=0..k-1} (x+j), and risefac(x,0) = 1. See the Hardy reference, pp. 104, 111.
The Morley formula gives the value of this series for |m| < 2/3 as Gamma(1-3*m/2)/(Gamma(1-m/2)^3)*cos(Pi*m/2). For the present case m=1/2 this value is hypergeometric([1/2,1/2,1/2],[1,1],1) = Pi/Gamma(3/4)^4 given in A091670.
REFERENCES
G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 104.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..555
F. Morley, On the Series 1 + (p/1)^3 + {p*(p+1)/1.2}^3 + ... , Proc. London Math. Soc. 34 (1902) 397-402, eq. (5), p. 401.
Eric Weisstein's World of Mathematics, Morley's Formula.
FORMULA
EXAMPLE
The rationals r(n) begin: 1, 9/8, 603/512, 4949/4096, 2576763/2097152, 20864151/16777216, 1347632055/1073741824, ...
The limit is given in A091670, approximately 1.3932039296856768591...
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Nov 11 2016
STATUS
approved