OFFSET
0,1
COMMENTS
This is the value of hypergeometric([1/4,1/4],[1],-1)^2. See A278143/A241756 for the partial sums of the hypergeometric series hypergeometric([1/2/,1/2,1/2],[1,1],-1) which has this value due to Clausen's formula. See the Hardy reference, p. 106, eq. (7.4.4) where this value is written as (Gamma(9/8)/(Gamma(5/4)*Gamma(7/8)))^2.
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 34.
G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 106, eq. (7.4.4)
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000
FORMULA
Equals hypergeometric([1/2/,1/2,1/2],[1,1],-1) = hypergeometric([1/4,1/4],[1],-1)^2 = Sum_{k>=0} (-1)^k*(risefac(k,1/2)/k!)^3, where risefac(x,m) = Product_{j =0..m-1} (x+j), and risefac(x,0) = 1.
Equals (Gamma(9/8)/(Gamma(5/4)*Gamma(7/8)))^2 = (sqrt(Pi)/(2^(1/4)*Gamma(5/8)*Gamma(7/8)))^2.
Equals Sum_{k>=0} (-1)^k * binomial(2*k,k)^3/64^k. - Amiram Eldar, Jul 04 2023
Equals Gamma(1/8)^4 * (2 - sqrt(2)) / (16 * Pi^2 * Gamma(1/4)^2). - Vaclav Kotesovec, Jul 04 2023
EXAMPLE
The value of the series 1 - (1/2)^3 + (1*3/(2*4))^3 - (1*3*5/(2*4*6) + ... is 0.909172794546929700739778854282651225720527299592205228386414021837...
This is also the value of the series Sum_{n>=0} c(n) with c(n) = Sum_{k=0..n} f(k)*f(n-k), where f(0)=1 and f(k) = (-1)^k*(1*5*9 *** (4*k-3)/(4*8*12 *** (4*k)))^2, k >= 1 (self-convolution of the hypergeometric([1/4,1/4],[1],-1) series).
MATHEMATICA
RealDigits[(Pi/Sqrt[2])*(1/(Gamma[5/8]*Gamma[7/8]))^2, 10, 50][[1]] (* G. C. Greubel, Jan 12 2017 *)
PROG
(PARI) (sqrt(Pi)/(2^(1/4)*gamma(5/8)*gamma(7/8)))^2 \\ Felix Fröhlich, Nov 15 2016
(Magma) pi:=Pi(RealField(110)); (Sqrt(pi)/(2^(1/4)*Gamma(5/8)*Gamma(7/8)))^2; // Felix Fröhlich, Nov 15 2016
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Wolfdieter Lang, Nov 14 2016
STATUS
approved