A068467 Decimal expansion of (1/4)! = Gamma(5/4).

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

Offset: 0

Author

Benoit Cloitre, Mar 10 2002

Keywords

• cons
• easy
• nonn

Examples

			0.906402477055477077982671288966918000748791920720...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• J. M. Borwein and I. J. Zucker, Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind, IMA Journal of Numerical Analysis, vol. 12, no. 4, pp. 519-526, 1992.
• Greg Martin, A product of Gamma function values at fractions with the same denominator, arXiv:0907.4384v1 [math.CA], 24-July-2009
• Albert Nijenhuis, Small Gamma Products with Simple Values, arXiv:0907.1689v1 [math.CA], 9-July-2009.
• Raimundas Vidunas, Expressions for values of the gamma function, arXiv:math/0403510 [math.CA], 30-March-2004.
• Wikipedia, Particular values of the Gamma function
• Index to sequences related to the Gamma function

Crossrefs

Cf. A202623.

Programs

Magma

SetDefaultRealField(RealField(100)); Gamma(5/4); // G. C. Greubel, Mar 11 2018

Maple

evalf(GAMMA(5/4)) ; # R. J. Mathar, Jan 10 2013

Mathematica

RealDigits[Gamma[5/4],10,120][[1]] (* Harvey P. Dale, Aug 23 2013 *)

PARI

gamma(5/4) \\ Altug Alkan, Sep 18 2016

Formula

2^(3/4)*(2/e^(16*Pi) + 1)* Pi^(3/4)/(2^(13/16)/(sqrt(2) - 1)^(1/4) + 2^(1/4) + 1) is a very good approximation (~88 digits) which becomes exact if you replace (2/e^(16*Pi) + 1) by EllipticTheta[3,0,exp(-(16*Pi))]. [R. W. Gosper, Posting to Math Fun Mailing List, Dec 27 2011.]

Equals A068466 /4 . - R. J. Mathar, Jan 10 2013

Also equals integral_{0..oo} exp(-x^4) dx. - Jean-François Alcover, Mar 29 2013

Equals 2^(-5/4)*Pi^(3/4)*Product_{k>=1} tanh(Pi*k/2). - Keshav Raghavan, Aug 25 2016

Extensions

Removed leading zero and adjusted offset, R. J. Mathar, Feb 06 2009

Additional reference from Joerg Arndt, Dec 28 2011

Edited by N. J. A. Sloane, Dec 28 2011