A202623 Decimal expansion of (1/3)! = Gamma(4/3).

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

Offset

0,1

Links

Table of n, a(n) for n=0..149.

Index to sequences related to the Gamma function

Formula

A formula from R. W. Gosper, Posting to Math Fun Mailing List, Dec 27 2011:

Equals (1/3) * (2*2^(7/9)*((Pi*EllipticTheta[3, 0, E^(-((16*Pi)/Sqrt[3]))])/ (1 + 1/(2^(1/4)*Sqrt[1 + Sqrt[3]]) + (2^(7/16)*((-1 + Sqrt[2])/(-Sqrt[2] + Sqrt[3]))^(1/4))/(-1+Sqrt[3])^(1/8)))^(2/3))/3^(1/4).

Equals Integral_{0..oo} exp(-x^3) dx. [Jean-François Alcover, Mar 29 2013]

Equals A073005/3. - R. J. Mathar, Jan 15 2021

Equals 3*Integral_{-1/e..0} (-LambertW(-1,x))^(1/3)-(-LambertW(x))^(1/3) dx. - Gleb Koloskov, Jun 07 2021

Example

0.89297951156924921121856431365822588137622979265243370031680...

Maple

evalf(GAMMA(4/3)) ;

Mathematica

RealDigits[(1/3)!, 10, 150][[1]] (* or *) RealDigits[Gamma[4/3], 10, 150] [[1]] (* Harvey P. Dale, Sep 03 2016 *)

Prog

(Macsyma)
4^(8/9)*%PI^(2/3)*THETA[3](0, %E^-(16*%PI/SQRT(3)))^(2/3)/(3^(1/4)*(2^(7/16)*(SQRT(2)-1)^(1/4)/((SQRT(3)-1)^(1/8)*(SQRT(3)-SQRT(2))^(1/4))+1/(2^(1/4)*SQRT(SQRT(3)+1))+1)^(2/3))
/* This is exact, but degrades to 50+ digits if you replace
THETA[3](0, %E^-(16*%PI/SQRT(3)))
by 1+2*%E^-(16*%PI/SQRT(3)) */
/* R. W. Gosper, Posting to Math Fun Mailing List, Dec 27 2011 */

Crossrefs

Cf. A068467, A073005.

Sequence in context: A242972 A198142 A336056 * A266261 A117914 A203077

Adjacent sequences: A202620 A202621 A202622 * A202624 A202625 A202626

Keyword

nonn,cons

Author

N. J. A. Sloane, Dec 29 2011

Extensions

Corrected and extended by Harvey P. Dale, Sep 03 2016

Status

approved