A327995 - OEIS

%I #19 Feb 04 2024 03:26:36

%S 9,2,0,4,4,1,7,8,7,8,3,5,5,9,0,9,8,3,9,3,4,9,1,7,1,3,0,7,4,9,9,9,0,9,

%T 8,1,2,2,9,4,9,8,9,2,0,2,0,9,1,5,1,3,4,2,2,5,3,3,0,0,0,5,8,9,1,8,1,5,

%U 3,5,0,3,7,4,1,5,1,3,1,3,4,3,5,5,4,9

%N Decimal expansion of Gamma(3/4)/Pi^(1/4).

%C The function df(x) = 2^(x/2)*(2/Pi)^(sin(Pi*x/2)^2/2)*Gamma(x/2+1) interpolates the double factorials A006882 and extends them analytically. df(-1/2) is the given constant. Extending also the notation this can be written as (-1/2)!! = (-1/4)!/Pi^(1/4).

%H Jean-Paul Allouche, Samin Riasat, and Jeffrey Shallit, <a href="https://doi.org/10.1007/s11139-017-9981-7">More infinite products: Thue-Morse and the Gamma function</a>, The Ramanujan Journal, Vol. 49 (2019), pp. 115-128; <a href="https://arxiv.org/abs/1709.03398">arXiv preprint</a>, arXiv:1709.03398 [math.NT], 2017.

%F Equals (-1/4)!/Pi^(1/4).

%F From _Amiram Eldar_, Feb 04 2024: (Start)

%F Equals Pi^(3/4)*sqrt(2)/Gamma(1/4).

%F Equals Product_{k>=0} ((4*k+1)*(4*k+4)/((4*k+2)*(4*k+3)))^A010060(k) (Allouche et al., 2019). (End)

%e 0.92044178783559098393491713074999098122949892...

%p Digits := 100: GAMMA(3/4)/Pi^(1/4)*10^86:

%p ListTools:-Reverse(convert(floor(%), base, 10));

%t RealDigits[Gamma[3/4]/Surd[Pi,4],10,120][[1]] (* _Harvey P. Dale_, Jun 13 2020 *)

%o (PARI) gamma(3/4)/Pi^(1/4) \\ _Michel Marcus_, Oct 24 2019

%Y Cf. A010060, A006882, A327996.

%K nonn,cons

%O 0,1

%A _Peter Luschny_, Oct 24 2019