cp's OEIS Frontend

A175574 Decimal expansion of sqrt(Pi) / (Gamma(3/4))^2.

%I A175574 #66 May 07 2025 16:56:09
%S A175574 1,1,8,0,3,4,0,5,9,9,0,1,6,0,9,6,2,2,6,0,4,5,3,3,7,9,4,0,5,5,8,4,8,8,
%T A175574 5,8,7,2,3,3,7,1,6,6,3,4,8,8,1,4,4,7,2,9,9,5,1,5,8,6,4,3,9,9,4,0,4,3,
%U A175574 0,4,1,8,0,7,2,0,7,1,5,7,9,4,9,7,8,4,5,8,6,1,6,1,9,5,8,0,7,9,5,4,2,0,9,4,5
%N A175574 Decimal expansion of sqrt(Pi) / (Gamma(3/4))^2.
%C A175574 Entry 34 c of chapter 11 of Ramanujan's second notebook.
%C A175574 This constant is also the ratio T(Pi/2)/T(0), where T(Pi/2) is the exact pendulum period for an amplitude of Pi/2 and T(0) the approximate period 2*Pi*sqrt(L/g) for small angles. - _Jean-François Alcover_, Aug 05 2014
%H A175574 G. C. Greubel, <a href="/A175574/b175574.txt">Table of n, a(n) for n = 1..5000</a>
%H A175574 Bruce C. Berndt, <a href="http://dx.doi.org/10.1112/blms/15.4.273">Chapter 11 of Ramanujan's second notebook</a>, Bull. Lond. Math. Soc., Vol. 15, No. 4 (1983), 273-320.
%H A175574 Claudio Carvalhaes and Patrick Suppes, <a href="https://doi.org/10.1119/1.2968864">Approximations for the period of the simple pendulum based on the arithmetic-geometric mean</a>, American Journal of Physics 76 (2008), 1150-1154.
%F A175574 Equals A002161 /A068465^2.
%F A175574 Equals 2F1([1/2,1/2],[1],1/2) = 1/agm(1, sqrt(1/2)) = gamma(1/4)^2/(2*Pi^(3/2)).
%F A175574 Equals 2*sqrt(2)*K(-1)/Pi, where K is the complete elliptic integral of the first kind, K(-1) being A085565. - _Jean-François Alcover_, Jun 03 2014
%F A175574 Equals Product_{k>=1} (1-(-1)^k/(2*k)) =  3/2 * 3/4 * 7/6 * 7/8 * 11/10 * 11/12 * ... .  - _Richard R. Forberg_, Dec 05 2015
%F A175574 Reciprocal of A096427. Equals ( Sum_{n = -inf..inf} exp(-Pi*n^2) )^2, a rapidly converging series. For example, summing from n = -5 to n = 5 gives the constant correct to 49 decimal places. - _Peter Bala_, Mar 06 2019
%F A175574 Equals Sum_{k>=0} binomial(2*k,k)^2/2^(5*k). - _Amiram Eldar_, Aug 26 2020
%F A175574 Equals (3/2)*hypergeom([-1/4, 3/4], [3/2], 1). - _Peter Bala_, Mar 04 2022
%F A175574 Equals A175573^2. - _Amiram Eldar_, Jul 04 2023
%e A175574 1.18034059901609622604533794..
%p A175574 sqrt(Pi)/GAMMA(3/4)^2 ; evalf(%) ;
%t A175574 First@ RealDigits[N[Sqrt@ Pi/Gamma[3/4]^2, 120]] (* _Michael De Vlieger_, Dec 06 2015 *)
%o A175574 (PARI) sqrt(Pi)/gamma(3/4)^2 \\ _Altug Alkan_, Dec 05 2015
%o A175574 (MATLAB) sqrt(pi)/gamma(3/4)^2 % _Altug Alkan_, Dec 05 2015
%Y A175574 Cf. A002161, A068465, A085565, A096427, A175573.
%K A175574 cons,easy,nonn
%O A175574 1,3
%A A175574 _R. J. Mathar_, Jul 15 2010
%E A175574 A-number typo for sqrt(Pi) corrected by _R. J. Mathar_, Aug 01 2010