A278140 - OEIS

%I #15 Jul 26 2022 22:03:14

%S 1,27,29835,914095,30845936835,966228811317,1005862016542383,

%T 31766194302634935,33673399154070922824435,1067731823813513897297545,

%U 1101976780048026596318593989,35023352480137647877041347193,1154564397329013014999165944225975

%N Numerator of partial sums of a Ramanujan series with value 2/(Gamma(3/4)^4), given in A277235.

%C The denominators are given in A074800.

%C 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.

%D G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 7, 105-106.

%F 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.

%F For (1+4*k)*((2*k-1)!!/(2*k)!!)^5 see A074799(k)/A074800(k).

%e The rationals r(n) begin: 1, 27/32, 29835/32768, 914095/1048576, 30845936835/34359738368, 966228811317/1099511627776, 1005862016542383/1125899906842624, ...

%e The limit r(n), for n -> oo, is 2/(Gamma(3/4)^4) given in A277235.

%Y Cf. A074799/A074800, A277235.

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_, Nov 13 2016