A327996 -id:A327996 - cp's OEIS frontend

A224268 Decimal expansion of Product_{n>=1} (1 - 1/(4n+1)^2).

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

Offset: 0

Bruno Berselli, Apr 02 2013

			0.9270373386506859592169251735976300231087994117608834527929640225280...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 34.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Peter Bala, Notes on the constants A096427 and A224268.

Cf. A064853, A085565, A093341, A327996.

Cf. product(1-1/(4n+r)^2, n>=1): A096427 (r=-1), A112628 (r=0), A179587-1 (r=2).

RealDigits[N[Product[1 - 1/(4 n + 1)^2, {n, 1, Infinity}], 90]][[1]] (* or, by the formula: *) RealDigits[Gamma[1/4]^2/(8 Sqrt[Pi]), 10, 90][[1]]

prodnumrat(1 - 1/(4*n+1)^2, 1) \\ Charles R Greathouse IV, Feb 07 2025

Equals Gamma(1/4)^2/(8*sqrt(Pi)) = L/(4*sqrt(2)), where L is the Lemniscate constant (A064853).
From Peter Bala, Feb 26 2019: (Start)
C = (Pi/4)*( Sum_{n = -inf..inf} exp(-Pi*n^2) )^2.
C = (-1)^m*2^(2*m+1)/Catalan(m) * Product_{n >= 1} ( 1 - (4*m + 3)^2/(4*n + 1)^2 ), for m = 0,1,2,....
C = Integral_{x = 0..1} 1/sqrt(1 + x^4) dx.
C = (1/sqrt(2))*Integral_{x = 0..1} 1/sqrt(1 - x^4) dx.
C = (3/2)*Integral_{x = 0..1} sqrt(1 + x^4) dx - sqrt(2)/2.
C = (1/8)*Integral_{x = 0..1} 1/(x - x^2)^(3/4) dx.
C = Sum_{n >= 0} binomial(-1/2,n)/(4*n + 1) = Sum_{n >= 0} binomial(2*n,n)/4^n * 1/(4*n + 1).
C = (1/2)*Sum_{n >= 0} (-1)^n*binomial(-3/4,n)/(4*n + 1).
Continued fraction: 1 - 1/(5 + 20/(1 + 30/(3 + ... + (4*n)*(4*n + 1)/(1 + (4*n + 1)*(4*n + 2)/(3 + ... ))))).
C = A085565/sqrt(2). C = Pi/(4*A096427). (End)
Equals A093341/2 = A327996^2. - Hugo Pfoertner, Oct 31 2024

A327995 Decimal expansion of Gamma(3/4)/Pi^(1/4).

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Oct 24 2019

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).

			0.92044178783559098393491713074999098122949892...

Jean-Paul Allouche, Samin Riasat, and Jeffrey Shallit, More infinite products: Thue-Morse and the Gamma function, The Ramanujan Journal, Vol. 49 (2019), pp. 115-128; arXiv preprint, arXiv:1709.03398 [math.NT], 2017.

Cf. A010060, A006882, A327996.

Digits := 100: GAMMA(3/4)/Pi^(1/4)*10^86:
ListTools:-Reverse(convert(floor(%), base, 10));

RealDigits[Gamma[3/4]/Surd[Pi,4],10,120][[1]] (* Harvey P. Dale, Jun 13 2020 *)

gamma(3/4)/Pi^(1/4) \\ Michel Marcus, Oct 24 2019

Equals (-1/4)!/Pi^(1/4).
From Amiram Eldar, Feb 04 2024: (Start)
Equals Pi^(3/4)*sqrt(2)/Gamma(1/4).
Equals Product_{k>=0} ((4*k+1)*(4*k+4)/((4*k+2)*(4*k+3)))^A010060(k) (Allouche et al., 2019). (End)

A319332 Decimal expansion of 1/2 + Sum_{n>0} exp(-Pi*n^2).

Original entry on oeis.org

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

Offset: 0

Hugo Pfoertner, Sep 18 2018

A part of Ramanujan's question 629 in the Journal of the Indian Mathematical Society (VII, 40) asked "... deduce the following: 1/2 + Sum_{n>=1} exp(-Pi*n^2) = sqrt(5*sqrt(5)-10) * (1/2 + Sum_{n>=1} exp(-5*Pi*n^2))."

			0.54321740560665400728765806075511172853510285362260944296039515799...

B. C. Berndt, Y. S. Choi, and S. Y. Kang, The problems submitted by Ramanujan to the Journal of Indian Math. Soc., in: Continued fractions, Contemporary Math., 236 (1999), 15-56, DOI: 10.1090/conm/236 (see Q629, JIMS VII).
B. C. Berndt, Y. S. Choi, and S. Y. Kang, The problems submitted by Ramanujan to the Journal of Indian Math. Soc., in: Continued fractions, Contemporary Math., 236 (1999), 15-56 (see Q629, JIMS VII).
Dan Romik, The Taylor coefficients of the Jacobi theta_3, arXiv:1807.06130 [math.NT], 2018.

Cf. A175573, A327996.

RealDigits[Pi^(1/4)/(2*Gamma[3/4]), 10, 120][[1]] (* Amiram Eldar, May 30 2023 *)

PARI
```
1/2+suminf(n=1,exp(-Pi*n*n))
```

sqrt(5*sqrt(5)-10)*(1/2+suminf(n=1,exp(-5*Pi*n*n)))

Equals Pi^(1/4)/(2*Gamma(3/4)). - Peter Luschny, Jun 11 2020
From Amiram Eldar, May 30 2023: (Start)
Equals Gamma(1/4)/(2*sqrt(2)*Pi^(3/4)).
Equals A327996 / sqrt(Pi). (End)

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A224268 Decimal expansion of Product_{n>=1} (1 - 1/(4n+1)^2).

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A327995 Decimal expansion of Gamma(3/4)/Pi^(1/4).

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A319332 Decimal expansion of 1/2 + Sum_{n>0} exp(-Pi*n^2).

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