OFFSET
0,2
COMMENTS
The denominators are given in A074800.
One of Ramanujan's series is 1 - 5*(1/2)^5 + 9*(1*3/(2*4))^5 - 13*(1*3*5/(2*4*6))^5 +- ... = Sum_{k>=0} (-1)^k*(1+4*k)*(risefac(1/2,k)/k!)^5 where risefac(x,k) = Product_{j =0..k-1} (x+j), and risefac(x,0) = 1. See the Hardy reference, p. 7, eq. (1.4) and pp. 105-106, 111. The value of this series is 2/(Gamma(3/4)^4) given in A277235.
REFERENCES
G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 7, 105-106.
FORMULA
a(n) = numerator(r(n)), with the rationals r(n) = Sum_{k=0..n} (-1)^k*(1+4*k)*(risefac(1/2,k)/k!)^5 = Sum_{k=0..n} (1+4*k)*(binomial(-1/2,k))^5 = Sum_{k=0..n} (-1)^k*(1+4*k)*((2*k-1)!!/(2*k)!!)^5. The rising factorial has been defined in a comment above. The double factorials are given in A001147 and A000165 with (-1)!! := 1.
EXAMPLE
The rationals r(n) begin: 1, 27/32, 29835/32768, 914095/1048576, 30845936835/34359738368, 966228811317/1099511627776, 1005862016542383/1125899906842624, ...
The limit r(n), for n -> oo, is 2/(Gamma(3/4)^4) given in A277235.
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Nov 13 2016
STATUS
approved