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

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, 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, 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

Offset: 1

Author

R. J. Mathar, Jul 15 2010

Keywords

• cons
• easy
• nonn

Comments

Entry 34 c of chapter 11 of Ramanujan's second notebook.

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

Examples

			1.18034059901609622604533794..
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Bruce C. Berndt, Chapter 11 of Ramanujan's second notebook, Bull. Lond. Math. Soc., Vol. 15, No. 4 (1983), 273-320.
• Claudio Carvalhaes and Patrick Suppes, Approximations for the period of the simple pendulum based on the arithmetic-geometric mean, American Journal of Physics 76 (2008), 1150-1154.

Crossrefs

Cf. A002161, A068465, A085565, A096427, A175573.

Programs

MATLAB

sqrt(pi)/gamma(3/4)^2 % Altug Alkan, Dec 05 2015

Maple

sqrt(Pi)/GAMMA(3/4)^2 ; evalf(%) ;

Mathematica

First@ RealDigits[N[Sqrt@ Pi/Gamma[3/4]^2, 120]] (* Michael De Vlieger, Dec 06 2015 *)

PARI

sqrt(Pi)/gamma(3/4)^2 \\ Altug Alkan, Dec 05 2015

Formula

Equals A002161 /A068465^2.

Equals 2F1([1/2,1/2],[1],1/2) = 1/agm(1, sqrt(1/2)) = gamma(1/4)^2/(2*Pi^(3/2)).

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

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

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

Equals Sum_{k>=0} binomial(2*k,k)^2/2^(5*k). - Amiram Eldar, Aug 26 2020

Equals (3/2)*hypergeom([-1/4, 3/4], [3/2], 1). - Peter Bala, Mar 04 2022

Equals A175573^2. - Amiram Eldar, Jul 04 2023

Extensions

A-number typo for sqrt(Pi) corrected by R. J. Mathar, Aug 01 2010