A278146 - OEIS

%I #12 Jan 10 2017 08:06:38

%S 1,0,6,2,6,7,9,8,9,9,9,1,6,8,4,3,6,5,1,1,8,2,4,9,0,1,9,5,1,0,4,5,1,2,

%T 0,9,1,0,6,2,5,4,9,9,1,8,3,2,6,0,2,0,6,9,4,2,4,1,0,5,4,8,7,4,0,7,3,3,

%U 9,6,1,1,1,2,7,1,8,2,2,8,3,6,7,4,0,2,9,9,0,9,3,7,2,0,4,0,6,3,7,4,5,8,6,7

%N Decimal expansion of 2^(3/2) / (sqrt(Pi)*Gamma(3/4)^2).

%C This is the value of a series of Ramanujan, namely 1 + 9*(1/4)^4 + 17*(1*5/(4*8))^4 + 25*(1*5*9/(4*8*12))^4 + ... = Sum_{k>=0} (1+8*k)*(risefac(1/4,k)/k!)^4 where risefac(x,k) = Product_{j=0..k-1} (x+j), and risefac(x,0) = 1. See the Hardy reference, p. 7, eq. (1.3) and p. 105, eq. (7.4.3) for s=1/4 (after division by s).

%C For the partial sums of this series see A278141/A278142.

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

%H G. C. Greubel, <a href="/A278146/b278146.txt">Table of n, a(n) for n = 1..5000</a>

%F 2^(3/2) / (sqrt(Pi)*Gamma(3/4)^2).

%e 1.06267989991684365118249019510...

%t First@ RealDigits@ N[2^(3/2)/(Sqrt[Pi] Gamma[3/4]^2), 104] (* _Michael De Vlieger_, Nov 15 2016 *)

%t RealDigits[2^(3/2)/(Sqrt[Pi]*(Gamma[3/4])^2), 10, 50][[1]] (* _G. C. Greubel_, Jan 10 2017 *)

%o (PARI) 2^(3/2)/(sqrt(Pi)*(gamma(3/4))^2) \\ _G. C. Greubel_, Jan 10 2017

%Y Cf. A278141/A278142.

%K nonn,cons

%O 1,3

%A _Wolfdieter Lang_, Nov 15 2016