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

%I A202623 #33 Jun 15 2021 11:04:49
%S A202623 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,
%T A202623 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,
%U A202623 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
%N A202623 Decimal expansion of (1/3)! = Gamma(4/3).
%H A202623 <a href="/index/Ga#gamma_function">Index to sequences related to the Gamma function</a>
%F A202623 A formula from R. W. Gosper, Posting to Math Fun Mailing List, Dec 27 2011:
%F A202623 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).
%F A202623 Equals Integral_{0..oo} exp(-x^3) dx. [_Jean-François Alcover_, Mar 29 2013]
%F A202623 Equals A073005/3. - _R. J. Mathar_, Jan 15 2021
%F A202623 Equals 3*Integral_{-1/e..0} (-LambertW(-1,x))^(1/3)-(-LambertW(x))^(1/3) dx. - _Gleb Koloskov_, Jun 07 2021
%e A202623 0.89297951156924921121856431365822588137622979265243370031680...
%p A202623 evalf(GAMMA(4/3)) ;
%t A202623 RealDigits[(1/3)!,10,150][[1]] (* or *) RealDigits[Gamma[4/3],10,150] [[1]] (* _Harvey P. Dale_, Sep 03 2016 *)
%o A202623 (Macsyma)
%o A202623 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))
%o A202623 /* This is exact, but degrades to 50+ digits if you replace
%o A202623 THETA[3](0,%E^-(16*%PI/SQRT(3)))
%o A202623 by 1+2*%E^-(16*%PI/SQRT(3)) */
%o A202623 /* R. W. Gosper, Posting to Math Fun Mailing List, Dec 27 2011 */
%Y A202623 Cf. A068467, A073005.
%K A202623 nonn,cons
%O A202623 0,1
%A A202623 _N. J. A. Sloane_, Dec 29 2011
%E A202623 Corrected and extended by _Harvey P. Dale_, Sep 03 2016